I don't fully understand how I can choose a submatrix in a matrix. Judging from this definition and picture (http://mathworld.wolfram.com/Submatrix.html), I would assume that you can't pick as a submatrix \begin{pmatrix}a_{1 1} & a_{1 2} \\ a_{4 1}&a_{4 2}\end{pmatrix} or \begin{pmatrix}a_{1 1} & a_{1 2} \\ a_{4 3}&a_{4 4}\end{pmatrix}, but is that true? More in general, my question is:
can you give a clear complete definition of submatrix and explain precisely how do I pick a submatrix?
Note: I'm mostly interested in this for the purpose of finding a non-zero determinant submatrix for calculating the rank of a matrix.
According to your link a submatrix is just a block of another matrix. So this is just a connected 'rectangle' of numbers of the original matrix, that means (as you already assumed) that you cannot skip rows or columns.
But if you consider following link, you can skip rows or columns, that means you can construct any submatrix by deleting whole columns and rows of the original matrix.
https://www.proofwiki.org/wiki/Definition:Submatrix
When looking at determinants, you probably mean the minors. These are submatrices that you get by deleting exaclty one row and one column of the original (square) matrix
https://en.wikipedia.org/wiki/Minor_%28linear_algebra%29