Reduce[eqns, vars] simplifies the equations eqns, attempting to solve for the variables vars. The equations generated by Reduce are equivalent to eqns, and contain all the possible solutions.
Reduce[eqns, vars, elims] simplifies the equations, trying to eliminate the variables elims.
The equations given to Reduce are in the form lhs == rhs. Simultaneous equations can either be given in a list, or combined with &&.
Example: Reduce[a x + b == 0, x] .
Reduce generates equations (==) and nonequalities (!=), combined with && and ||.
Reduce primarily deals with polynomial equations.
See The Mathematica Book: Section 1.5.7 and Section 3.4.6.
See also: Solve, Eliminate, LogicalExpand, ToRules.