Consider Mount Elbrus to be a winter climb needing basic alpine skills on a non-technical but snow-covered slope at altitude. You will need crampons, a walking axe, a helmet, mountaineering boots, a harness with slings and karabiners to clip into a rope, a proper mountain down jacket and down mitts as well as your layers of fleece and shell. See the Preparing for Elbrus page for the full kit list and more information on preparing for a mountain expedition.
Crampons are needed once you go above the snowline and a walking axe is necessary for self arrest and walking poles with snow baskets are also a must. A climbing helmet is recommended; the danger is not from falling rock, but a head injury occurring during a slip or fall. A harness with locking carabiners and a sling is necessary so you can clip into the fixed-line above the Saddle, and we would recommend an alpine harness because it is easier and safer to put on with boots and crampons.
The guides will carry ropes in case of an emergency or if they feel that someone would benefit from being roped up, particularly on the descent.
For north route expeditions, you will additionally need a four-season sleeping bag (a three-season bag would suffice on the south side because the huts can be quite warm) and a sleeping mat but the equipment otherwise is much the same.