We have said that we would like to perform elementary operations in our matrix to get simpler ones. But what does simpler mean? Which is the ideal situation we should pursue?

A matrix is said to be in row echelon form if

1. All zero rows (rows containing only $0$) are below the nonzero ones
2. The first nonzero coefficient in a nonzero row (called leading coefficient or pivot) is $1$
3. Each pivot is strictly to the right of the pivot of the row above it

A matrix is said to be in reduced row echelon form if moreover

1. All coefficients above a pivot are $0$