General linear group ($GL(n)$, $GL(n,F)$)

Invertible matrices. Or if you think a bit more generally, an invertible linear map.

When the field $F$ is not given, it defaults to the real numbers.

Non-invertible are excluded "because" otherwise it would not form a group (every element must have an inverse). This is therefore the largest possible group under matrix multiplication, other matrix multiplication groups being subgroups of it.

general linear group over a finite field of order $m$. Remember that due to the classification of finite fields, there is one single field for each prime power $m$.

Exactly as over the real numbers, you just put the finite field elements into a $n\times n$ matrix, and then take the invertible ones.