A vacuum pump is just an air pump that you use for sucking rather than blowing. You can convert an old-fashioned "floor pump" ("bike pump") into a vacuum pump that can take over 75 percent of the air out of something. That's several times as strong as a vacuum cleaner's pull, and three quarters as strong as the best possible vacuum pump.
The net atmospheric force on whatever you evacuate will be over 11 pounds per square inch (or over 1500 pounds per square foot), which is enough for many tasks requiring vacuum.
(For those of you who are not metrically impaired like most of us in the U.S., that's over 7700 kilograms per square meter.)
(1) Vacuum pumps made this way are in use around the world, in non-industrialized areas, for vacuum packing food. (Especially to save seeds for the next year's planting, by killing bugs and preventing germination. See http://www.plenty.org/soy/vacuumpump.html
(2) A manual vacuum pump can used to make a "vacuum press" to clamp laminates together with a ton or more of force while the glue sets. (A smaller manual pump, designed for evacuating wine bottles, is often used for laminating skateboard decks, using a commercial vacuum bagging kit from Roarockit called the "Thin Air Press". Our bigger pump pulls about as hard, but requires considerably less pumping.) It can also be used for vacuum bagging composites like fiberglass, to squeeze out excess resin and make stronger, lighter parts.
(3) You can use it with a vacuum former, to form thicker plastic sheets than you could with just a vacuum cleaner. (Doug Walsh shows how to apply this to vacuum forming in his book "Do It Yourself Vacuum Forming for the Hobbyist," available from http://www.build-stuff.com
(4) You can implode various things with it, or attach it to a vacuum chamber and expand or explode things in the chamber; that's often fun and/or instructive. (Try Peeps.)
The vacuum generated is not strong enough for some tasks, like evacuating refrigerant systems or degassing difficult-to-degas liquids. For those tasks you need a pump with more leverage, that takes almost all of the air out.
A plain bike pump is just a piston pump rather like a syringe, with a rubbery disk sliding in a cylinder. Drawing back the disk sucks air into the cylinder, and pushing it in pushes air out.
To make this syringe-like thing work as a pump, sucking from one place and blowing to another, two "check valves" (one-way valves) are used. One lets it suck in from an air intake, but not blow out the intake. The other lets it blow out the exhaust, but not suck in from the exhaust.
Unlike most piston pumps, a bike pump uses the rubbery disk both as a piston and as one of the check valves. When you pull back on the pump, the disk flexes inward and allows air around it, into the cylinder ahead of the piston. When you push inward, the disk is stopped from flexing the other way by a metal disk (like a big washer); its edge seals against the inside of the cylinder, so that when you push it down, it compresses the air in the cylinder and forces it out the exhaust.
(There's a normal check valve at the exhaust, to keep it from sucking the air right back in on the upstroke, and more air flows in behind the piston through a hole in the cylinder top.)
To convert the pump, we'll need to do two things:
(1) Reverse the piston disk and the metal disk that backs it up, so that it seals on the upstroke (to create vacuum) and flexes on the downstroke (to let air around it and out of the cylinder).
(2) "Reverse" the exhaust check valve, so that it lets air in but not back out, and we can use the old exhaust connection as the new air intake. Actually, it's usually easier just to remove the check valve, and replace it with one that does the right thing, so that's what we'll do.
To make it easy to do both of these things, we'll want a simple, cheap, old-fashioned bike pump with no frills (like a pressure gauge) to complicate things.
(If you want a small electric vacuum pump, have a look at my other instructable on converting a 12V "tire inflator" air compressor: https://www.instructables.com/id/E791HNXF23Z39P6/