Basic Information
The assault rifle is a selective-fire rifle designed to give the user the widest possible range of capabilities in combat. Some authorities divide the class further into the battle rifle which fires a 'full power' round (such as the H&K G3 or the FN-FAL, which fire 7.62mmNATO) and the remainder (which retain the assault rifle designation) which fire lower power ammunition, but this is not a universally recognised distinction and is further complicated by the fact that there is no set standard for 'full power' rifle rounds1. The selective fire may be between semi-automatic-action and any one of several other options - normally fully-automatic-action or a burst of three rounds, but others are possible. Some more complicated designs may have several options.
Generally the weapon combines the capability to generate large volumes of fire in close combat (originally the task of a submachinegun) with the ability to operate at a decent range (which was previously the province of self loading rifles or bolt action rifles) - hence the wider range of capabilities. It also tends to be smaller and lighter than its rifle ancestors - an important consideration in an era of increased mechanisation and troops with large varieties of other kit to carry about.
Some assault rifle designs will include a variant which is "optimised" for supporting fire and incorporates some of the features of the light machine gun - this design is often best classified2 as a support rifle, even if the issuer choses to call it a machine gun.
These days, the assault rifle is the standard armament of the modern infantryman (and most supporting troops as well). Most modern designs are in a 'low power' ammunition calibre3 and are optimised for an engagement range of up to 300m based on data retrieved from a several modern conflicts, including the Vietnam War and Korea, although these assumptions are under review as a result of the ongoing Afghan Conflict.
The ancestor (and 'trope namer') of the modern assault rifle was the sturmgewehr 1944 (a.k.a the Maschinenpistole 44), developed by the Germans during WW2 to improve the capabilites of their beleagured forces. The name was a political decision, calculated to attract the approval of a Fuhrer who was otherwise convinced that an automatic rifle would foster a defensive mindset amongst his troops, but the name stuck (in translation at least) and has been popular ever since.
The StG44 was not the first selective fire rifle ever made, nor yet the first to be issued at scale (the BAR and the earlier FG424 preceeded it by years), but it was the first to effectively combine selective fire with controlability and a light enough overall built to make it a sensible mass issue weapon. Mikhail Kalashnikov's 1947 rifle (yes, the AK-47) followed swiftly behind the StG44, and soon everyone was at it.
Modern assault rifles also tend to be fitted with a variety of useful accessories - and those used by PCs are particularly famous for being pimped out like this - the StG44 started the trend by being the first weapon to commonly mount an infra-red night sight (and the Krummlauf curved barrel, which was a lot less use) and modern rifles commonly mount accessories as diverse as laser designators, under barrel grenade launchers or shotguns and torches. The wide range of possible fits has started a trend of developing universal mounting systems (like the US Picatinny Rail - or "P-Rail" system) which can be installed on the weapon and then allow the user to mount and dismount kit without needing the assistance of an armourer.