Where a square root of a number "x" is some number "y" that, when multiplied by itself, gives "x", a (perfect) square of a number is the result of multiplying a number by itself. That is to say, the square of "y", by multiplying "y" by "y", is "x". You can also talk about "squaring" a number, which is to find what the square is. (It can be both a noun and a verb.)
For example: the square of 4 is "4 x 4" = 16. Also, if you square 4, you get 16.
If you want to think of a visual representation of it, "what is the square of 5" is essentially the same as asking "what is the area of a square with a length of 5?" (Of course, all sides have equal lengths in a square.) As you can probably figure out already, to find the area of a square (or, in general, a rectangle) you multiply the length by the width. So here, it is obviously 5 x 5 and the area, or the square of 5, is 25.
This concept of perfect squares can also be extended to polynomials. For example, let's look at the following case:
(x+1)*(x+1) is the square of (x+1). It can also be written as (x+1)^2. You can do the visual trick i just described above if you want, using x+1 as the side length.
If you multiply (FOIL) these binomials, you get (x^2 + 2x + 1). As it is equal to our original expression, you can also say that this product is a perfect square, just as you can say that 16 or 25 is a perfect square. To find out what the square root of this is expression is, it is the same as asking what the square root of (x+1)*(x+1). Over time you will see patterns and be able to quickly notice that the square root of (x^2 + 2x + 1) is (x+1).
This shouldn't be too difficult of a concept to understand. I will do a few short examples, but post comments if you require clarification:
Find the square of 12:
12 x 12 = 144
Find the square of 25:
25 x 25 = 625
Find the square of (x-1):
(x-1)*(x-1) = (x-1)^2 = (x^2 - 2x + 1)
Find the square of (2x+3):
(2x+3)*(2x+3) = (2x+3)^2 = (4x^2 + 12x + 9)
It can also be noticed, and should be kept in mind, that squares will always be positive. Try it to see: plus x plus = plus... negative x negative = plus. A plus times a negative is NOT a square! Squares multiply the SAME number (sign and all!).
See my previous post about "completing the square" for some more stuff relevant to this post.