Having been so much improved by the purchase of a pair of Puma Replicats, as compared to my previous New Balance cross-training shoes, I have taken a great interest in what everyone else is wearing at the gym ("box"). I list here what seem to be the top 6 most popular shoes for "Elite-Fitness":
Note: be sure to read my updated article, The Latest and Greatest "Elite-Fitness" Shoes.
Sambas are lightweight indoor soccer shoes with flat soles and almost no heels. Because they are designed for indoor soccer, they provide great lateral support and balance. Their soles are just a little padded and they do have some minimal arch support. They are great for everything at "Elite-Fitness", even running, as long as you use don't heel-strike (See the POSE running technique). They cost about $45.
Ontisuka Tigers are very popular around the box here, it seems like almost a third of us own them. They are flat soled shoes with hardly any differential between the toe and heel. Depending on what type of Ontisuka Tiger you get, the heels will have more or less padding. With more padding you'll get much greater comfort and they'll absorb more impact when running and doing double-unders; however, it's a slight trade-off because they'll be a little mushier for the lifts. The Mexico 66 Ontisuka Tiger is the model with a flatter heel, while the Ultimate 81 Ontisuka Tiger has a slightly more raised heel. They both cost about $65.
I mention these shoes because so many people in the "box" own them and wear them to the workouts (no trainers, though). The Nike Free running shoes try to simulate the barefoot experience while still providing lots of padding for your feet. The soles of the shoes, which are rather thick, are divided into a bunch of square sections that are able to move an bend apart from eachother. This means that when you step on something on the right side of the ball of your foot, the pressure won't be pulling on the rest of your foot, so the right side will have to bear its own weight, almost as if you were barefoot. This allows you to have a thick mushy sole but still exercise the muscles in your foot. However, if you run correctly without heel striking (as in the POSE method) then you don't need a mushy sole because there isn't much impact anyways. If you have a thin sole like in the Tigers or Sambas then you'll still get the barefoot experience and don't need to divide up the soles into squares.
The Frees are pretty expensive ($85) and are really meant for running. Apparently they can get torn up doing rope-climbs. Since their soles are so thick they would not be ideal for weight-lifting. If you do get some Frees, I would get the Free 3.0's because they have the flattest soles.
Vibrams Five Fingers
These shoes are really minimalistic foot coverings. They basically provide a layer of rubber over the sole your foot and that's about all. They are as close to barefoot running as you can get with a shoe without actually being barefoot. Running and exercising barefooted is beneficial to developing the muscles in your feet, and actually reduces the risk of joint and muscle injuries in the legs.
Vibram Five Fingers (VFF) are great for almost any "Elite-Fitness" workout, even the running WOD's. There are people who run marathons barefooted; you just have to do it right. The problem with VFFs is that you can't wear a sock with them so they smell bad quickly, but they can be washed easily enough in the washing machine. Also, they look very strange. The soles of VFFs are thick enough to protect your feet from pebbles and maybe shards of glass, but they are thin enough that its still uncomfortable to run on gravel.
Inov-8 F-Lite 230
This shoe has a fairly flat sole with gnarly grips. It is ideal for trail running. They are extremely light weight, and are good for everything in "Elite-Fitness". I actually haven't seen a pair of these in real-life (we can't afford them around here) so I can't say much about them, but they are popular and highly recommended by others. They cost about $105. Although the 230's are often the ones specifically recommended on forums, the more expensive F-Lite 195's have a more neutral balance between toe and heel, so they ought to be better.
Converse Chuck Taylor All-Stars
These little shoes are classic and inexpensive ($30 or so). They really are minimalistic, with just a canvas top and a thin rubber sole. They are not especially light weight, considering their minimalism. Beyond providing some shoe support, they also give the wearer a sense of pride that he/she is partaking in the same classic style experienced by millions since their conception in the 1910's.
I hope that helps you decide which shoe to wear. I myself wear Puma Repli Cat III's, which are almost perfectly flat and have pretty much no padding or arch support, but are too narrow and have no lateral support. If I were to buy a shoe now, I'd probably get Sambas or Tiger Mexico 66's.