Can I use electrical tape for making a splice? If so, how?
Edit- Based on comments above tape cannot be used (alone) for creating a splice. As Tester101 pointed out if the splice was soldered or crimped or joined with some (UL approved) device, then tape may be used for insulating the splice. 

With regard to splices made with connectors, the method below could be used for insulating the connector (of specifically, a pigtail type of splice).

There are at least three kinds of electrical tape that I am aware of: vinyl, rubber, and mastic. For this purpose I would use rubber and vinyl, (reserving mastic for split bolt or similar connectors). I'm not sure if you must use rubber, but it will provide more thickness and (I believe) better moisture protection than vinyl. The tape used should be UL listed and approved for insulating splices. Generally speaking, the insulating tape should be 1.5 times thicker than the conductor's original insulation (source- http://ecmweb.com/content/electrical-taping-skills-lost-art).

If the connector has an exposed metal surface then I would start by wrapping it with (at least) one layer of mastic tape. Rubber tape will be used next (supposing the connector was not insulated), then vinyl tape on top of the rubber tape. Rubber and vinyl tape will be wrapped around the splice using a process called half lapping. Half lapping tape means to cover 1/2 of the tape every time you go around it. Start by stretching the rubber tape until it almost snaps and tear or cut off the end. Begin wrapping about 1.5 inches below the connector or stripped part of the wires. Keep tension on the tape (stretching it until it is ready to snap) while wrapping. When you get to the end of the splice, continue to wrap the tape upon itself two more times (wrapping the tape around itself), then fold the tape down, and wrap over it, continuing the half lap process back to where you started; cover the area where taping began and then (changing direction) continue wrapping back toward the end of the splice. Repeat the wrapping process, but this time fold the tape over the opposite direction (and wrap back down to where the tape was started). Cut the rubber tape and get the vinyl tape ready. 

Rubber tape should be wrapped over the splice at least 4 times: up = 1 wrap, fold, down = 2 wraps, up = 3, and back down = 4. Then vinyl tape should be wrapped over that (again, using half laps four times). 

Start by exposing a few inches of the vinyl tape. Pull it so that it stretched a little (about 30% length). Cut the end off and begin wrapping just below the rubber tape. Use the same half-lapping method, keeping tension on the tape so that it is stretching (slightly) while wrapping. When you are done the diameter of the splice should be thick enough (about 10 mm thick around the two wires); but if not, you should wrap it once more with vinyl tape (up and down) to be sure.