The soldier is in effect the default class; a soldier has no other purpose other than killing the enemy. As a result, soldiers can select from the full range of weapons -- including the more exotic weapons such as Mausers, panzerfausts, or Venoms. The compensation for having no special abilities is that soldiers start with a considerable amount of ammunition.
Play a solder when you don't feel like doing anything other than killing, or when you're starting out and aren't familiar with the character classes.
The primary purpose of the engineer is to plant and arm dynamite. This is used to achieve objectives; often one side has to breach a wall or gate before continuing. Engineers must first plant the dynamite (which emits a yellow glow), and then arm it with the pliers (after which it glows blinking red) -- two steps that each expose them to enemy fire. Engineers also have the ability to disarm dynamic planted by the enemy, which is essential in foiling the assaulting enemy. Finally, engineers can repair machine gun emplacements that have been damaged. Engineers have the restricted set of weapons to choose from, like the other specialized classes, but to compensate they start with more grenades than the other classes.
The engineer is an essential part of the team. On any objective-based game, you need to have at least a few engineers on your team who know what they're doing.
Medics do just what they sound like -- they help other players. Medics can do two essential things: First, they can drop down health packs which will boost the health of anyone (including the enemy) who picks them up. A medic's health slowly regenerates over time, and additionally, when a team starts with medics, all players on the team spawn with 10 more health points. More importantly, though, medics can revive downed friendlies (with the syringe), provided they haven't abandoned to limbo mode. Medics are perhaps the most essential possible class; a team without good medics is doomed.
Medics must be attentive to their teammates and quick with the syringe.
Lieutenants are really a multipurpose class. Their immediate purpose is to drop down ammunition packets for their teammates (but which can be picked up by the enemy for the same purpose as well). In addition, they have two very powerful strategic abilities. First, they can, call in an air strike (only to area exposed to the sky) by throwing a special multicolored smoke grenade; within a few seconds, the sound of aircraft can be heard and the area around the multicolored smoke is haphazardly battered with what amounts to napalm. Second, with his binoculars, a lieutenant can call in a tank barrage; again, though, the area must be exposed. A smaller, bluish smoke grenade is spontaneously launched at the remote site, and a few seconds later sporadic fire from a nearby tank unit pounds the area. Both of these abilities are not very effective in short-range tactical combat, but can be extremely useful in strategic operations -- clearing an area before an incursion, or as suppressive fire for a tight area.
The ability to replenish ammunition for teammates is powerfully important, but at the same calling in air strikes and tank barrages is extremely effective for keeping the enemy at bay.