A flow proof is just one representational style for the logical steps that go into proving a theorem or other proposition; rather than progress downward in two columns, as traditional proofs do, flow proofs utilize boxes and linking arrows to show the structure of the argument. All of the theorems, properties, definitions and postulates that support each step appear next to the boxes.
The flowchart structure of this type of proof is quite similar to the diagrams that computer programmers often use when putting together their lines of code. Flow proofs work well for geometric as well as algebraic proofs, making the steps and their rationales easier for one's audience to understand.
The two-column proof is the traditional one used in high school geometry books, and it is also known as the ledger proof or the T-form proof. The shape, givens and task appear at the top of the page, but the columns "Statements" and "Reasons" take the reader through the proof step by step.
Those pursuing geometry further into college often use paragraph proofs. This is a thorough paragraph that takes the reader through the entire process of the proof, one sentence at a time. Without the structure of the columns or the flowchart, it is easy to forget steps or reasons, so it is wise to take care when making this sort of proof.