Perhaps Kanil can give us a bit more information, I'd also be happy to hear about it.
Can't guarantee this is accurate, but basically each faction has build tables. When you buy a unit, it rolls and compares to the build table. This lets LC get Commandos, while DC gets Jenners and FS Valkyries, etc.
In Solaris, you get 8 'mechs from the Solaris build tables at random (I believe it's 4 lights, 3 mediums, and a heavy? Could have changed!) and after every battle you can reset your 8 'mechs. Or keep them for the duration of Solaris. It's basically the training faction. Your 'mechs don't take permanent damage or die, so you can lose them freely. A typical game is lance v lance (4v4.)
Once you choose to leave Solaris, you get the same 4/3/1, but this time from your faction's tables, so you don't keep your Solaris 'mechs. These new 'mechs can (and will) get damaged and die (except fighting players still in Solaris), and replacing them costs c-bills, which are earned from fighting. Generally players will surrender once they feel they cannot win in order to preserve their remaining units, lowering their repair/replacement costs. Your opponent doesn't have to accept this surrender, but it's quite rare for them to decline.
If you're careless, you can run out of 'mechs, but it's pretty hard to do. Especially considering that, since 'mech production is random, there's usually a few 'mechs people are literally willing to give away to you. 'Mechs require support (termed "bays"), and sometimes you get something from the factory that it's just not worth keeping to you (but might be useful to someone else, depending upon their playstyle. UrbanMechs, for example.)
Factions have a pool of units (some donated by players to free up bays, the others autoproduced by factories with resource surpluses) that you can buy from at a discount compared to a new unit from the factory. Only catch is that you only will get (at random, of course) what's already in the surplus pool. It's generally a good idea to buy these things, if what's in the pool is worth using, since they're discounted.
Kind of broken as fuck explanation of a few concepts, but if you have any specific questions, feel free to post them or PM me and I'll try my best to explain. But, most importantly, while you're in Solaris you cannot fuck anything up, so just have fun.