A penny saved is a penny

Haskell, 37 34 bytes

s#l@(c:d)|s>=c=(s-c)#l+s#d
s#_=0^s

Usage example: 26 # [1,5,10,25] -> 13.

Simple recursive approach: try both the next number in the list (as long as it is less or equal to the amount) and skip it. If subtracting the number leads to an amount of zero, take a 1 else (or if the list runs out of elements) take a 0. Sum those 1s and 0s.

Edit: @Damien: saved 3 bytes by pointing to a shorter base case for the recursion (which also can be found in @xnors answer).

Mathematica, 35 22 bytes

Thanks to miles for suggesting FrobeniusSolve and saving 13 bytes.

Length@*FrobeniusSolve

Evaluates to an unnamed function, which takes the list of coins as the first argument and the target value as the second. FrobeniusSolve is a shorthand for solving Diophantine equations of the form

a1x1 + a2x2 + ... + anxn = b

for the xi over the non-negative integers and gives us all the solutions.

Jelly (fork), 2 bytes

æf

This relies on a branch of Jelly where I was working on implementing Frobenius solve atoms so unfortunately you cannot try it online.

Usage

$ ./jelly eun 'æf' '12' '[1,5,10]'
4
$ ./jelly eun 'æf' '26' '[1,5,10,25]'
13
$ ./jelly eun 'æf' '19' '[2,7,12]'
2
$ ./jelly eun 'æf' '13' '[2,8,25]'
0

Explanation

æf  Input: total T, denominations D
æf  Frobenius count, determines the number of solutions
    of nonnegative X such that X dot-product D = T