I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change f:Z\to Y, meaning that X \times_{Y} Z \to Z also has this property.
Question: Is the base change in number theory and derived algebraic geometry the same thing as above? What would be the examples?
In number theory, base change can also refer to an operation on automorphic representations. If L/K is an extension of number fields, and pi is an automorphic representation of a reductive group G over K, then pi should "lift" to a new automorphic representation of G over L. This is the sense of the phrase used in, e.g., Langlands' book "Base Change for GL(2)". The existence of a certain kind of base change for GL(2) was used to prove the modularity of some mod 3 Galois representations, which in turn played a role in proving Fermat's last theorem.
In number theory, base change refers to tensor product: the operation in the category of rings corresponding to fibred product in the category of (affine) schemes.
So, if A is a k-algebra, and K is a field extension of k (or less typically, another k-algebra), then the "base change of A to K" refers to A \otimes_k K.
(I would imagine in derived algebraic geometry it refers to a fibred product as usual, though I'm not sure.)
In number theory, base change of a scheme or a variety is with respect to the underlying ring or field, is viewing the same scheme/variety over an extended ring or field, but with the "same" set of equations.
For example given a curve over $\mathbb Q$, it is also a curve over any number field. Or given a scheme over spectrum of $\mathbb Z$ given by some equation, you can reduce it modulo a prime $p$ and obtain a scheme over $\mathbb F_p$.
When you are dealing with group schemes, moduli, motives, etc., such notions carry over through base change, modulo technical details.
