There are many acceptable methods for making a wire splice and you need to consider many different factors when making a splice. Here are a few considerations.

• High Voltage vs. Low Voltage – If the connection is 24V or less it USUALLY has fewer NEC (National Electrical Code) rules and regulations about how the connections are made and in some cases you are safe making an inline splice without a box. When making an inline splice on high voltage conductors you MUST use a properly UL rate splice or a box.
• Dry vs. Damp Conditions – If it's ever going to be exposed to moisture you need to think about shorting and corrosion. Your splice should keep water away from the conductors themselves. if there is any chance of moisture.
• Concealed vs. Accessible – If you are going to bury the splice in the ground or in a wall it needs to be RATED for that purpose and you need to be darn sure that splice will last as long as the conductor itself.
• Quality of Connection – Every connection needs to be good, but in cases like communication or AV wires it needs to be PERFECT. Think of that new high efficiency, super fancy communicating HVAC system you are installing. Those comm connections need to be good.    
• Tension the Connection is (or may be) Under – In other words is the wire stretched or is there a chance it might be stretched or pulled later. For example, if a splice is going to be pulled inside a conduit, there is a good chance it will be pulled out someday. If the next guy tries to pull it out and it comes apart, your name will be cursed.
• Aesthetics  – If the splice looks like a hunk of junk, it will be assumed it is a hunk of junk by everyone who sees it. Neat workmanship matters.

Here are a few options for splicing wires depending on application –

Bad Options 

Splicing any high voltage conductor in an “open” manner or in way that is not specifically rated. In most cases get a UL rated connector and make the connection inside a UL / NEMA rated rated box or assembly.

Making a splice by just twisting wires together and putting electrical tape on top. Just don't.

Using wire nuts and creating a big ball of wires and running electrical tape over them until it looks like a giant blob of tape.

Good Options

Use wire nuts on low voltage or control wire in dry and accessible conditions but twist them so the wires stay neat and lay half of the conductors in one direction and the other half in the other direction and tape up in a neat fashion.

The same type of configuration with 3M Scotchlok crimp connectors for better moisture resistance than wire nuts.

In some mildly damp conditions you may be able to use self fusing silicone tape for a more water resistant layer than electrical tape.

Use butt end connectors on stranded wire or if using small gauge single conductor wire (like 18ga stat wire) you can double the end of the wire over before making a crimp. When making a crimp ensure that that the actual crimp is made on the side of the connector OPPOSITE  the seam. Once you make a butt end connector pull HARD on it totect and ensure that no bare wire is exposed outside of the insulator.

Better Options 

Use heat shrink butt connectors and stagger the connections to reduce the bulge. Heat the connectors to seal them, then run a piece heat shrink over them all. I found this 4:1 shrink ratio, marine grade heat shrink that should do a great job or water proofing. Heat shrink can be a real life saver and you can use a heat gun or a small butane torch to heat it up. Coincidentally they also make little, portable butane soldering irons as well.

For better connection quality and strength… that's when you may consider the fabled NASA Splice!

When making a soldered splice make sure to use rosin core solder and wipe off the rosin flux before covering the splice to help prevent corrosion. Remember to run the heat shrink over the cable and the individual conductors BEFORE you start making the splices to prevent sadness and yelling.

Best Options

The best options are to just run a new wire or make the connections inside of a rate box with proper connectors. Sometimes the best way is the simplest way.

— Bryan