Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.
The words "process" and "finite number of operations" have been taken to mean that Hilbert was asking for an algorithm. The term "rational integer" simply refers to the integers, positive, negative or zero: $0,pm\; 1,pm\; 2,\; ldots$. So Hilbert was asking for a general algorithm to decide whether a given polynomial Diophantine equation with integer coefficients has a solution in integers. The answer to the problem is now known to be in the negative: no such general algorithm can exist. Although it is unlikely that Hilbert had conceived of such a possibility, before going on to list the problems, he did presciently remark:
"Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses or in the sense contemplated."
The work on the problem has been in terms of solutions in natural numbers rather than arbitrary integers. But it is easy to see that an algorithm in either case could be used to obtain one for the other. If one possessed an algorithm to determine solvability in natural numbers, it could be used to determine whether an equation in $n$ unknowns,
has an integer solution by applying the supposed algorithm to the $2^n$ equations
Conversely, an algorithm to test for solvability in arbitrary integers could be used to test a given equation for solvability in natural numbers by applying that supposed algorithm to the equation obtained from the given equation by replacing each unknown by the sum of the squares of four new unknowns. This works because of Lagrange's foursquare theorem, to the effect that every natural number can be written as the sum of four squares.
Sets of natural numbers, of pairs of natural numbers (or even of $n$tuples of natural numbers) that have Diophantine definitions are called Diophantine sets. Diophantine definitions can be provided by simultaneous systems of equations as well as by individual equations because the system
is equivalent to the single equation
A recursively enumerable set can be characterized as one for which there exists an algorithm that will ultimately halt when a member of the set is provided as input, but will continue indefinitely when the input is a non member. It was the development of computability theory (also known as recursion theory) that provided a precise explication of the intutitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous. It is evident that Diophantine sets are recursively enumerable. This is because one can arrange all possible tuples of values of the unknowns in a sequence and then, for a given value of the parameter(s), test these tuples, one after another, to see whether they are solutions of the corresponding equation. The unsolvability of Hilbert's tenth problem is a consequence of the surprising fact that the converse is true:
Every recursively enumerable set is Diophantine.This result is variously known as Matiyasevich's theorem (because he provided the crucial step that completed the proof) and the MRDP theorem (for Yuri Matiyasevich, Julia Robinson, Martin Davis, and Hilary Putnam). Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate consequence. In fact, more can be said: there is a polynomial
Year  Events 

1944  Emil Leon Post declares that Hilbert's tenth problem "begs for an unsolvability proof". 
1949  Martin Davis uses Kurt Gödel's method for applying the Chinese Remainder Theorem as a coding trick to obtain his normal form for recursively enumerable sets:

1950  Julia Robinson, unaware of Davis's work, but grasping the key importance of the exponential function, attempts to prove that EXP, the set of triplets

1959  Working together Davis and Putnam study exponential Diophantine sets, that is, sets definable by Diophantine equations in which some of the exponents may be unknowns. Using the Davis normal form together with Robinson's methods, but assuming the then unproved conjecture that there are arbitrarily long arithmetic progressions consisting of prime numbers, they prove that every recursively enumerable set is exponential Diophantine, and as a consequence, that J.R. implies that every recursively enumerable set is Diophantine, and that Hilbert's tenth problem is unsolvable. 
1960  Robinson shows how to avoid the unproved conjecture about primes in arithmetic progression and then greatly simplifies the proof. J.R. is revealed as the key to further progress, though many doubt that it is true. 
19611969  Davis and Putnam find various propositions that imply J.R. Matiyasevich publishes some reductions of Hilbert's tenth problem. Robinson shows that the existence of an infinite Diophantine set of primes would suffice to establish J.R. 
1970  Matiyasevich exhibits a system of 10 simultaneous first and second degree equations which provide a Diophantine definition of the set of pairs $(a,b)$ such that

The Matiyasevich/MRDP Theorem relates two notions — one from computability theory, the other from number theory — and has some surprising consequences. Perhaps the most surprising is the existence of a universal Diophantine equation:
This can be seen to be true simply because there are universal Turing machines, capable of executing any algorithm. This implication of J.R. had made it seem implausible.
Hilary Putnam has pointed out that for any Diophantine set $S$ of positive integers, there is a polynomial
such that $S$ consists of exactly the positive numbers among the values assumed by $q$ as the variables
range over all natural numbers. This can be seen as follows: If
provides a Diophantine definition of $S$, then it suffices to set
So, for example, there is a polynomial for which the positive part of its range is exactly the prime numbers. (On the other hand no polynomial can only take on prime values.)
Other applications concern what logicians refer to as $Pi^\{0\}\_1$ propositions, sometimes also called propositions of Goldbach type. These are like the Goldbach Conjecture, in stating that all natural numbers possess a certain property that is algorithmically checkable for each particular number. The Matiyasevich/MRDP Theorem implies that each such proposition is equivalent to a statement that asserts that some particular Diophantine equation has no solutions in natural numbers. A number of important and celebrated problems are of this form: in particular, Fermat's Last Theorem, the Riemann Hypothesis, and the Four Color Theorem. In addition the assertion that particular formal systems such as Peano Arithmetic or ZFC are consistent can be expressed as $Pi^\{0\}\_1$ sentences. The idea is to follow Kurt Gödel in coding proofs by natural numbers in such a way that the property of being the number representing a proof is algorithmically checkable.
$Pi^\{0\}\_1$ sentences have the special property that if they are false, that fact will be provable in any of the usual formal systems. This is because the falsity amounts to the existence of a counterexample which can be verified by simple arithmetic. So if a $Pi^\{0\}\_1$ sentence is such that neither it nor its negation is provable in one of these systems, that sentence must be true.
A particularly striking form of Gödel's incompleteness theorem is also a consequence of the Matiyasevich/MRDP Theorem:
Let
provide a Diophantine definition of a noncomputable set. Let $A$ be an algorithm that outputs a sequence of natural numbers $n$ such that the corresponding equation
has no solutions in natural numbers. Then there is a number $n\_0$ which is not output by $A$ while in fact the equation
has no solutions in natural numbers.
To see that the theorem is true, it suffices to notice that if there were no such number $n\_0$, one could algorithmically test membership of a number $n$ in this noncomputable set by simultaneously running the algorithm $A$ to see whether $n$ is output while also checking all possible $k$tuples of natural numbers seeking a solution of the equation
We may associate an algorithm $A$ with any of the usual formal systems such as Peano Arithmetic or ZFC by letting it systematically generate consequences of the axioms and then output a number $n$ whenever a sentence of the form
is generated. Then the theorem tells us that either a false statement of this form is proved or a true one remains unproved in the system in question.
We may speak of the degree of a Diophantine set as being the least degree of a polynomial in an equation defining that set. Similarly, we can call the dimension of such a set the least number of unknowns in a defining equation. Because of the existence of a universal Diophantine equation, it is clear that there are absolute upper bounds to both of these quantities, and there has been much interest in determining these bounds.
Already in the 1920s Thoralf Skolem showed that any Diophantine equation is equivalent to one of degree 4 or less. His trick was to introduce new unknowns by equations setting them equal to the square of an unknown or the product of two unknowns. Repetition of this process results in a system of second degree equations; then an equation of degree 4 is obtained by summing the squares. So every Diophantine set is trivially of degree 4 or less. It is not known whether this result is best possible.
Julia Robinson and Yuri Matiyasevich showed that every Diophantine set has dimension no greater than 13. Later, Matiyasevich sharpened their methods to show that 9 unknowns suffice. Although no one imagines that this striking result is the best possible, there has been no further progress. So, in particular, there is no algorithm for testing Diophantine equations with 9 or fewer unknowns for solvability in natural numbers. For the case of rational integer solutions (as Hilbert had originally posed it), the 4 squares trick shows that there is no algorithm for equations with no more than 36 unknowns. But Zhi Wei Sun showed that the problem for integers is unsolvable even for equations with no more than 11 unknowns.
Martin Davis studied algorithmic questions involving the number of solutions of a Diophantine equation. Hilbert's tenth problem asks whether or not that number is 0. Let $A=\{0,1,2,3,ldots,aleph\_0\}$ and let $C$ be a proper nonempty subset of $A$. Davis proved that there is no algorithm to test a given Diophantine equation to determine whether the number of its solutions is a member of the set $C$. Thus there is no algorithm to determine whether the number of solutions of a Diophantine equation is finite, odd, a perfect square, a prime, etc.
Although Hilbert posed the problem for the rational integers, it can obviously be just as well asked for many rings. Obvious examples are the rings of integers of algebraic number fields as well as the rational numbers. As Hilbert was surely aware, an algorithm such as he was requesting could have been extended to cover these other domains. For example, the equation
where $p$ is a polynomial of degree $d$ is solvable in nonnegative rational numbers if and only if
is solvable in natural numbers. (If one possessed an algorithm to determine solvability in nonnegative rational numbers, it could easily be used to determine solvability in the rationals.) However, knowing that there is no such algorithm as Hilbert had desired says nothing about these other domains.
There has been much work on Hilbert's tenth problem for the rings of integers of algebraic number fields. Basing themselves on earlier work by Jan Denef and Leonard Lipschitz and using class field theory, Harold N. Shapiro and Alexandra Shlapentokh were able to prove:
Hilbert's tenth problem is unsolvable for the ring of integers of any algebraic number field whose Galois group over the rationals is abelian.
Shlapentokh and Thanases Pheidas (independently of one another) obtained the same result for algebraic number fields admitting exactly one pair of complex conjugate embeddings.
The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open. Likewise, despite much interest, the problem for equations over the rationals remains open.