A money market account is a deposit account with a relatively high rate of interest, and short notice (or no notice) required for withdrawals. In the United States, it is a style of instant access deposit subject to federal savings account regulations, such as a monthly transaction limit.
In the United States, a money market deposit account is a deposit account that is considered a savings account for some purposes, but upon which checks can typically be written, subject to certain restrictions. Like a Negotiable Order of Withdrawal account, it is structured to comply with Regulation Q, which forbids paying interest on checking accounts. Thus money market deposit accounts are accounts that bear interest, and on which checks can be written, but, due to various restrictions, are not legally checking accounts, and thus do not run afoul of Regulation Q.
Typical restrictions are that a fairly high minimum balance must be maintained in order to avoid fees. With the advent of online banking, many banks are able to pay a high interest rate on a low balance, sometimes as low as $1. A debit card is often issued for making withdrawals.
In theory, the restrictions allow the bank to invest the money with more discretion, allowing a higher return. The return is often competitive with money market mutual funds, although nothing requires a bank to invest deposits in these types of accounts into the money market.
Although money market deposit accounts have a similar name to money market funds, they are not the same: a money market fund consists of assets held by a brokerage (or bank) on behalf of investors, while a money market deposit account is a deposit at the bank, and hence a liability of the bank towards depositors.
A money market fund is a kind of mutual fund (technically, a regulated investment company). Investors receive shares in this company, which buys securities (for example, commercial paper). There are rules on what kind of securities may be held and rules about diversification. Thus, investors have risk on the assets, but not on the bank.
A money market account is simply a liability of the bank (albeit a high-priority one). It is a note on the bank's books that it owes someone money. It has no specific assets; essentially, it is backed by the entire bank. Thus, investors have risk on the bank, but not (directly) on any assets that the bank may invest in with these deposits – in fact, the deposits will not in general match up with any particular assets: they are simply one among many liabilities of the bank.
Also, like a checking account, these accounts are insured by the FDIC or a state analog.