Modulus Question

[Alphanos]

Sorry if I sound stupid asking this, but what is meant by the expression "1 (mod p)", where p is greater than one? I'm sure it must make more sense than I can figure out.

If the context is important, I'm reading up on RSA, quote's here:

Let p and q be two different large primes. A large number p can be tested for primeness by applying Fermat's theorem, which states that if p is prime, then
a^p-1 == 1 (mod p)

for every positive integer a not divisible by p. (A double equal sign (==) is used instead of the usual symbol for congruence, which is not supported by all Web browsers.) A number p which passes this test for several hundred randomly selected values of a in the range (1, p-1) is almost surely prime.

[kedaman]

the rest after division with p, for instance 13=3 (mod 5)

[alkatran]

x mod y = z
3 mod 2 = 1
4 mod 2 = 0

X= its the amout between X and the nearest multiple of Y

[Shaggy Hiker]

Isn't 1 mod P where P>1 always going to be 1?

[bugzpodder]

for integers a,p,q, a>=0, p>1
q= a (mod p)

means q can be represented as k*p+a, where k is some integer, 0<=a<p