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A matrix is in row echelon form if non-zero rows appear before zero rows and each non-zero row has a leading (pivot) entry to the right of the pivot above. REF allows back-substitution to find solutions. Pivot columns identify basic variables; non-pivot columns identify free variables.
From your study of Gaussian elimination, you know the process: apply row operations to systematically eliminate unknowns from equations below each pivot row. Row echelon form (REF) is the name for the structured shape that results. Think of it as a descending staircase from left to right: each non-zero row has its first nonzero entry — the pivot — strictly to the right of the pivot in the row above, and any all-zero rows sink to the bottom. The matrix does not have to have zeros above the pivots; that extra cleanup produces the reduced row echelon form, which comes later.
The staircase shape makes solving the system mechanical through back substitution. Starting from the bottommost non-zero row, you have one equation involving one or a few unknowns. Solve for the leading variable. Substitute its value upward into the next row, again giving you one new equation with one new leading variable. Continue upward. Each step resolves one variable; by the time you reach the top row, every variable is determined. You never have to wrestle with two unknowns simultaneously — elimination already did that work.
The pivot positions tell you something fundamental about the solution structure. Columns with pivots correspond to basic variables — each has a unique value once free variables are assigned. Columns without pivots correspond to free variables — each can take any value, with basic variables adjusting to compensate. If there are r pivots across n unknowns, there are n − r free variables. A system with free variables has infinitely many solutions, forming a parameterized family. No solutions arise when a zero row is paired with a nonzero right-hand side — a contradiction of the form 0 = c ≠ 0.
A practical detail: two different sequences of row operations on the same matrix can produce different row echelon forms — REF is not unique. But the number of pivots and which columns contain them are invariant regardless of the path you take. That count of pivots is the rank of the matrix, and it has geometric meaning: it equals the dimension of the column space, the number of independent constraints the system actually imposes on the unknowns. Everything important about the solution structure is encoded in rank and the pivot/free-variable split.