Specifically, a matrix is in row echelon form if

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All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes (all zero rows, if any, belong at the bottom of the matrix).
The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.
All entries in a column below a leading entry are zeroes (implied by the first two criteria).^[1]

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Some texts add the condition that the leading coefficient of any nonzero row must be 1.^[2]

Tutorial Math 204 Tutorial 1 System of linear equations Gauss / Gauss - Jordan reduction