The kinds of expressions which can refer (as so defined) are:
(1) a noun phrase of any structure, such as: the taxi in The taxi's waiting outside; the apple on the table in Bring me the apple on the table; and those five boys in Those five boys were off school last week. In those languages which, like English, encode definiteness, REs are typically marked for definiteness. In the examples given, this is done by the definite article the or the demonstrative adjective, here those.
(2) a noun-phrase surrogate, i.e. a pronoun, such as it in It's waiting outside and Bring me it; and they in They were off school last week. The referent of such a pronoun may vary according to context - e.g. the referent of me depends on who the speaker is - and this property is technically an instance of deixis.
(3) a proper name, like Sarah, London, The Eiffel Tower, or The Beatles. The intimate link between proper names and type (1) REs is shown by the definite article that appears in many of them. In many languages this happens far more consistently than in English. Proper names are often taken to refer, in principle, to the same referent independently of the context in which the name is used and in all possible worlds, i.e. they are in Saul Kripke's terminology rigid designators.
Referring can take place in a number of ways. Typically, in the case of (1), the RE is likely to succeed in picking out the referent because the words in the expression and the way they are combined together give an true, accurate, description of the referent, in such a way that the hearer of the expression can recognize the speaker's intention. In the first example, if the hearer knows what an apple and a table are, and understands the relation expressed by on, and is aware that the is a signal that an individual thing/person is intended, s/he can build up the meaning of the expression from the words and grammar and use it to identify an intended object (often within sight, or at any rate easily recoverable, but not necessarily). Of course, the speaker may use a mistaken description and still manage to refer successfully. If I ask you to "Take this plate to the woman with the glass of vodka", you may take it to the intended person even if, unbeknown to me, her vodka is really water. On the other hand I may be accurate in calling it vodka, but you may believe wrongly that it is water, and therefore not deliver the plate. So accurate reference is not a guarantee of successful reference, and successful reference does not wholly depend on accurate reference. But naturally there is a strong positive correlation between them.
Proper names, on the other hand, generally achieve reference irrespective of the meaning of the words which constitute them (if any are recognizable). If my local pub is called The Anchor, this is simply a label which functions conversationally with no appeal to the meaning of the words. If I say, I'm going to the Anchor, I do not mean I'm going to the device for halting and securing a ship, and you will not necessarily call such a device to mind when I say this. The Anchor just serves to identify a particular building. This point is more obvious still with those names like Sarah and London which have no lexical meaning of their own.
In addition to the (in many languages) grammatically obvious singular and plural reference, linguists typically distinguish individual or specific reference, exemplified by each case presented so far, from generic reference, where a singular expression picks out a type of object (etc.) rather than an individual one, as in The bear is a dangerous animal. Plural expressions can, of course, be interpreted in the same way, as in Bears are dangerous animals.
Definite reference to single individuals is usually taken to be the prototypical type of reference.
Other types of reference recognized by linguists include indefinite as opposed to definite reference, and collective and distributive reference. Definite referring expressions refer to an identifiable individual or class (The Dalai Lama; The Coldstream Guards; the student with the highest marks), whilst indefinite referring expressions allow latitude in identifying the referent (a corrupt Member of Parliament; a cat with black ears - where a is to be interpreted as 'any' or 'some actual but unspecified'). Collective reference is the picking out of the members of a set as a set, whilst distributive reference is the picking out of the members of a set individually. The difference may not be marked linguistically, but arrived at by interpretation in context. Compare Manchester United won again today (where the reference of Manchester United is to members of the team as a unit), with Manchester United wear red shirts and black shorts (where the reference of Manchester United is to the team members as individuals). English allows such expressions to be ambiguous: compare Manchester United are rich beyond my wildest dreams.
Reference and denotation
This is an important distinction. Denotation is the relation existing between a lexical item and a set of potential referents in some world. Reference is the relation between some expression and actual referents (subject to the technical restriction given above). The word rabbit denotes the entire class of objects that are classified with this term, whilst the RE my rabbit will generally refer, on a particular occasion of usage, to the one individual in my possession. Generally speaking, lexical items have denotation, whilst phrases have the job of doing reference in real situations. This distinction is not systematically made by some linguists.
Some technical linguistic characteristics of referring expressions
REs carry a presupposition of the existence of the referent(s), in some universe of discourse, including fictional universes (in which Oliver Twist, Noddy, Daleks, Mansfield Park, Atlantis, purgatory, the man with the golden gun, phlogiston, and the invisible pink unicorn "exist").
There are many other technical issues surrounding the nature of reference. Some of these are discussed from the perspective of linguistics in Lyons (1977, vol. I: chapter 7); Cann (1993: chapters 9 and 10); Saeed (1997: chapters 2, 7, 11). There is a vast literature on the topic in philosophy.