When is it generally safe/acceptable to use a wire nut?
Wire nuts come in a lot of different sizes and styles, just look at this PDF.  This brochure is from Ideal and go to page 20 for the wire size chart.  I used information from Ideal not because I think they are better than others but I'm more familiar with them.

@BMitch is right about mixing AL &amp; CU, but there are wire nuts for those applications as well.  Most of these say that you CANNOT use with aluminum to aluminum.  Weird. 

While I don't think its a problem on this site, you can use most wire nuts with 600 volt maximum.  Normally the colors dictate what sizes you can use with what wires and a lot of times these will overlap one another.  Yellow, tan, red, and the large grey will allow you to use 2 - #12's with each.  Yellow allows 1 to 3 #12's while the large grey allows 2 to 6 # 12's.  The colors and the represented wire sizes accepted is pretty much accepted by most manufacturers.

Direct burial wire nuts are available.  They are 'gel' filled and have a thicker shell.  I don't know if I would direct bury one except for landscape lighting, but they are ideal for being used in damp, moist areas like pole lights.

Anything larger than the listed size should be used with split bolts, which gets treated completely different.