There are several adhesives that can hold pieces of wood together without screws or nails. The cheapest and most commonly available is plain old white glue, similar to the Elmer's we used in school. Officially called polyvinyl resin, white glue sets fairly quickly and becomes transparent when it hardens. It's convenient for gluing dowels in place on those ready-to-assemble furniture pieces and for similar woodworking projects.
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Next up the ladder is carpenter's glue, or aliphatic resin glue. Carpenter's glue is stronger than white glue, and sets faster. It also holds up better to sanding than white glue does. Any joining that uses either white glue or carpenter's glue should be clamped together until the glue sets.
Neither white nor carpenter's glue is suitable for outdoor use, or anywhere it will get wet. For those you'll want formaldehyde glue, also called resorcinol. Formaldehyde glue is good for doors, windows, and moldings that might be exposed to water.
By the way, all three of these glues are different from construction adhesives, which don't penetrate bare wood well and shouldn't be used for joining wood pieces. One exception is installing molding or trim in tight spaces where you can't use a clamp. Construction adhesive is sticky enough to hold wood trim without clamping.
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