If you've found parts of a vintage table at a flea market or you've dragged one up from the basement only to find it can't stand on its own -- don't get discouraged. There are plenty of ways to make a base for it. Go retro, modern or rustic with a variety of materials that might be available at the same place you found the table top.
Lots of pedestal tables rely on pyramid bases. This type of base looks just like the name implies, with graduating layers of hardwood plywood or solid lumber, each piece 3/4 inch thick. The height is determined by how many pieces you use. For example, three pieces will make a base 2 1/4 inches tall. Stack additional pieces together for a taller base. For even taller bases, stack two identical pieces of each size together. Start with a section approximately 18 inches square, and make each following section 1 inch smaller than its predecessor to create the pyramid effect. Add glue to each piece and nail them together in a stack. Screw the base to the bottom of the table. It's heavy and provides the needed ballast.
Plywood and Hardwood
Build a basic box base using hardwood and plywood. This type of base uses a square or rectangular footprint for stability, and you can make one that matches the existing woodwork in your home. Start with four pieces of 3/4-inch hardwood 4 inches wide. Cut them to length using a miter saw and stand them together on edge to create a frame that's approximately 18 inches square for a moderately sized table. If your table is large, make the base bigger or even rectangular. Cut a 1/2-inch-wide, 3/4-inch-deep rabbet or channel along the top edge using a table saw with a dado blade. Assemble the frame with nails or screws, with the rabbet facing up. Cut a piece of hardwood plywood to fit into the channel; add glue, and nail it in to serve as the top of the base. The base looks like a shallow box. Cut two more pieces of plywood and nail them inside the box for ballast. Screw the base to the bottom of the table.
Simple half-lap joints create a basic base of a type that's been used for many years. It consists of two pieces of lumber joined in the middle. Start with two pieces of 3/4-by-4-by-24-inch lumber for a moderately sized table. Install a 3/4-inch dado blade on a table saw. Using a miter gauge, cut identical slots 1/2-inch deep, by 4 inches wide, centered on each piece of lumber to create the half lap joint. Add glue and place the two channels or slots together with clamps. The provides a flat, durable cross-like member that screws to the bottom of the table to provide a large, unassuming footprint for the table. This type of base resembles the traditional Christmas tree stand nailed to the bottom of the tree.
Lots of things work well as table bases. Older, retro lamp bases look trendy and will often adapt to the bottom of a table. For a recreation room table, try some type of wheel or spindle. You can attach these types of bases to the table with drilled 1/8-inch steel plates or purchase some metal brackets from a home supply store. Use your imagination -- you might find that a base from an old television, radio or any other type of discarded furniture looks great as a base. Scavenge second hand stores, flea markets or yard sales -- you'll come up with something.