A field is a set with two binary operations called addition and multiplication satisfying various axioms. Wikipedia article: Field_(mathematics)
A field extension is when you add a new element and then have to add all arithmetic combinations of that new element with the existing elements, e.g. adding i to the real numbers to get the complex numbers. If F is a subfield of K then K is called an extension field of F.
Technically you aren't adding new elements, but defining a new field which has a subset isomorphic to the orginal field, e.g. the complex numbers x+0i are identified with the real numbers x.
Finite field extensions are useful in the construction of BCH codes which are codes over the finite fields of order q=pn which are extensions of the fields of order p where p is prime. These finite fields known as Galois fields GF(p) are extended to GF(q) by adding roots of polynomials that are irreducible (non-factorable) over GF(p).
Where else in CS does the concept of extension fields arise ? A fundamental appearance in some subject would be more interesting than an occurence in some proof, but if it is used frequently in a common proof stategy then such a proof strategy would also be an interesting answer here.