How many chain links do I need? Looking for chains at ebay, I see, that they have 106, 114, 116 links and so on. 

I have a racing bike (2x7) and see, that chains are often announced as 6-7-8-speed, 10-speed, 11-speed. Since there are some prime numbers, I guess it is only back gear, which is taken into account? 

I found one picture with a printing on the containment: "6-7-8 speed" and "116 Links". So am I right - would this thing be alright for me, or do I have to take more information into account? 

Manufacture of the bike is Peugeot, of the mechanics: Shimano exage, an 80ies bike, and the chain lost attachment very abruptly after the cold winter. I'm not stick to Shimano, if I don't need to. Location Germany, if this is of importance.
The speed is the number of cogs on the rear cassette. This matters because higher numbers of gears (esp 9speed) mean that the chain must be narrower to fit between them.
Generally 6-7-8 speed chains are the same and 9 speed are thinner, more than 9 speed is a bit specialized.

The length isn't as important because you will usually have to shorten the chain to fit your bike. The standard chain is 114 pins (56 links) but you almost never need to know this ( I had to look it up!).