Regiment is a military unit designation that goes back far into military history, but, in many modern forces, it has become a historical term with the more common equivalent being brigade. Traditionally, when one speaks of an army unit as the "3rd Infantry", "3rd Infantry Regiment" is implied when there is no other unit term, or the notation does not indicate a smaller unit (e.g., 1/3 Infantry would be, in U.S. usage, 1st battalion, 3rd Infantry Regiment).
Brigades and regiments are, in contemporary forces, units of roughly 3,000-5,000 personnel. Regiments are commanded by colonels; a brigade is often commanded by a colonel, but, especially when operating independently, may be commanded by a brigadier general.
In the U.S. Army, "regiment" is essentially a historical term. The 3rd Armored Cavalry Regiment is organized as a heavy brigade. While the 75th Ranger Regiment does exist as a real headquarters, Ranger operations usually are of battalion or smaller size. U.S. Marine Corps regiments do exist as training and administrative organizations, but, if an entire regiment is deployed, it will be as the ground forces component of a Marine Expeditionary Brigade.
British tradition was that a "regiment" was a home headquarters, which might train and recruit soldiers but have one or more battalions as operational units. A Royal Marine Commando is designated as a regiment.
Russia does use regiments as tactical organizations, which tend to be considerably smaller than a western brigade.