I'm going to start with one that isn't subject to much dispute.
Flow-through (also called pass-through) means an accelerated payment of certain subsidiary rights income.
When you sign a contract, you usually grant the publisher the primary right to print, publish, and sell your manuscript in book form. You also grant to the publisher certain subsidiary rights in the work, and most often the publisher will then try to license these rights to a third party. Once a sale is made, the publisher and author will split any money received.
What subsidiary rights you license and at what splits depends on the book, the publisher, and your agent, but for the purposes of this post I'm just going to discuss what happens when the publisher does make a subsidiary rights sale to a third party. Typically the author's share of the money made from the license is paid to the author in the next royalty statement, which could be up to a year away. A flow-through clause requires that the money be paid to the author within a certain period of time - i.e. that the money flows directly through to the author without the normal delays.
Typically a flow-through clause only applies once an author has earned out their advance. Also, publishers almost always require that the sale for the license be for more than a certain amount of money in order to justify the cost for the extra bookkeeping. Finally, I've found that the more subsidiary rights you grant the publisher, the more likely a publisher will agree to include this clause.