Any evaluation strategy where evaluation of some or all function arguments is started before their value is required. A typical example is call-by-value, where all arguments are passed evaluated. The opposite of eager evaluation is call-by-need where evaluation of an argument is only started when it is required.
The term "speculative evaluation" is very close in meaning to eager evaluation but is applied mostly to parallel architectures whereas eager evaluation is used of both sequential and parallel evaluators.
Eager evaluation does not specify exactly when argument evaluation takes place - it might be done fully speculatively (all redexes in the program reduced in parallel) or may be done by the caller just before the function is entered.
The term "eager evaluation" was invented by Carl Hewitt and Henry Baker <email@example.com> and used in their paper ["The Incremental Garbage Collection of Processes", Sigplan Notices, Aug 1977. (ftp://ftp.netcom.com/pub/hb/hbaker/Futures.html)]. It was named after their "eager beaver" evaluator.