No law requires partners to create a written partnership agreement, but it's smart to do so. If you don't make a partnership agreement, you run the risk that the default rules in your state's partnership laws will govern your partnership in ways you and your partners won't like.
Creating a written partnership agreement will also give you and your partners a chance to discuss your expectations of each other, define how each of you will participate in the business, and help you work out any sticky issues before they become major problems.
You don't have to spend a fortune on lawyer's fees to create a valid agreement -- you and your partners can easily put together a simple, clear agreement yourselves.
For more information, read Creating a Partnership Agreement.