Yes, it is usually a good idea to actually put any agreements with others into writing. And, yes, certain agreements--like an agreement to buy a house--must be in writing to be enforceable. But you can sue--or be sued--for breach of contract even if the agreement was never in writing.
A written agreement is actually one of three kinds of agreements that can be biding under law. A second kind is an oral agreement. Oral agreements, as the term suggests, are agreements that are made through spoken words. Asking the neighbor's kid to shovel your driveway for $50 might be an example of a binding oral contract.
The third kind of binding agreement is known as an implied contract. Implied contracts are made through conduct not words. Remember the squeegee men on the streets of New York City? They would rush to your car and start washing your window shield during red lights and then demand that you pay them for their service. Even though you had never asked them to wash your window shield, believe it or not, a binding implied contract could be argued to have been made by virtue of the fact that you sat there in your car watching the man perform the service and not objecting to his doing the work. (Of course, many would object but to no avail).
In the business world, the stakes are often far higher than the example of the squeegee man or the neighbor's kid. But the same concepts apply. There might be a binding agreement even if the agreement was not in writing. So a breach of contract can exist even if the agreement in question was made through spoken words or conduct.