A lattice is a partical ordered set (set with a partial order), where every subset containing 2 elements (subsets {a,b}) has a least upper bound (supremum, sup) and a greatest lower bound (infimum, inf).

A lower bound i is a infimum of a set S, iff for every x with for all y of S x <= y , x <= i holds. Dito for supremum with >=.

A lattice is called complete iff every subset has a supremum and an infimum.
A set whose all subsets {a,b} have a supremum is called upper half lattice. In a lower half lattice only the infimums must exist. Upper/lower half lattices are called complete iff the supremums/infimums of all subsets exist.

Examples: Set of subsets of a set is a complete lattice.
Boolean algebras are complete lattices (in fact lattices can be used, along with other constraints, to define a boolean algebra)
The real numbers are a lattice, but not complete.