An armchair is a chair with two side armrests. It can be a large, overstuffed recliner, a small wooden chair, or anything in between, but they're built for comfortable living or sitting room use.
Wing chairs are built with large, enclosing "wings" going down either side, originally as protection against drafty windows. Often used near fireplaces, wing chairs are usually associated with wealth and luxury.
Lounge chairs are built with relaxation in mind. These chairs are often deeply reclined into an almost supine position for maximum comfort.
Adjustable chairs are built to accomodate multiple positions for a versatile seating solution.