A Ring Set

White gold and diamond ring set
By EraGem

A ring set is a group of ring which are made to be worn and fit together, the most common set being an engagement and wedding ring set.

Engagement Ring

Platinum and diamond engagement ring
By Neil Lane
Oval cut diamond with microset smalls

An engagement ring, is commonly given in Western tradition by a man to a woman as a symbol for the commitment to be married. There is no specific type of ring which is classed as an engagement ring but most common are solitaires and clusters.

Eternity Ring

A set of claw set eternity bands
by Langerman diamonds
18 karat white gold and diamonds

An eternity ring is a ring which is encircled with identically sized stones, also known as a 'full eternity' it symbolic meaning is 'never ending love. Common setting for and eternity ring are claw, bead and channel.

Dress Ring

Gold and diamond Dress ring
by Sterlé from Morelle Davidson.  
circa 1970

A dress rings would be a ring that is not an engagement or wedding ring, worn only for occasions of on a semi-regular basis. As the word implies the  dress ring is worn for dressing-up.

Cluster Ring

Antique Diamond Cluster Ring
by Jewelry Atelier
18k yellow gold and platinum, .88 carat center Old European cut diamond and 1 carat of accent diamonds

A cluster setting contains a group of stones, traditionally a single large stone surrounded by smaller stones. 

Solitaire Ring

Diamond Solitaire Ring
by Randor Jewellery

A solitaire ring has a single stone as the focal point of the design