Electrical tape acts as an insulator over bare wiring and can be used for temporary repairs on worn or damaged insulation on light-gauge wires such as lamp cords, but it's not suitable for joining wires or as a protective cover for household circuit wires that are twisted together. Alternatives such as plastic wire connectors have replaced electrical tape for many home wiring connections. Never attempt repairs on live wires -- unplug freestanding devices or ensure the breaker is off for home repairs before working on electrical items.
Wire connectors, commonly called wire nuts, are plastic caps with internal threads. They twist onto the bare ends of wires to create a secure, insulated, yet reversible connection. Wire nuts have long been the standard devices for joining circuit wires in household wiring systems, while twisting wires together and covering the connection with electrical tape is no longer accepted by most building codes. Check the wire connector package to ensure you acquire the right size connector, as specifics vary based on the wire gauge as well as the number of wires being connected.
Heat-shrink tubing looks a bit like a straw or hollow plastic coffee stirrer, depending on its thickness. This tubing slides over one wire before it's soldered to another. After the bare wire ends are twisted and soldered together, slide the tubing over the wire connection. A heat source such as a heat gun shrinks the tubing tightly over the connection so the wires do not come apart. Heat-shrink tubing often is used on small wires in electronic devices but is seldom used in household wiring systems.