Why did they put this additional one ?
They did not "put" and an additional one. These are all examples of different number system: that is, a set of number along with operations you can do.
I thought real numbers and complex numbers together make up all the elements of a matrix.
No we can have more than just real matrices and complex matrices. Check Andrea's answer.
Also, you may want to check the Wikipedia page for Real numbers, complex numbers, integers, rational numbers and integer modulo prime.
What do they mean by integers modulo a prime ?
Integers modulo prime, often denoted as $\mathbb{Z}/p \Bbb{Z}$ where $p$ is a prime. The form a finite field, but this won't be helpful for you now. For now, you can think of it as the numbers in the range $\{0, 1, 2, \ldots, p-1 \},$ and all the arithmetic ($+,-,\times,\div$) are performed modulo prime.
They have many applications e.g. in cryptography, coding theory. That's why we often need to consider matrices with such entries. On the other hand, engineering and physics deal with real & complex numbers, so we need matrices over the reals and complex numbers. Different domains different applications.
If you want to know more, you can read the Wikipage on modular arithmetic.