A lot of different types of logos are available in the market. Some of the popular ones include:
Wordmarks : They are basically free standing words or an abbreviation of several words that forms a logo. Some of the popular companies like Google, IBM and eBay use this form of logo.
Pictorial : Such logos have picture of recognizable things. Companies like Twitter and Starbucks have such kind of logos. The picture is the brand identity.
Letterform : In this form, the logo design idea is to just have a single letter format. Companies like Uber, McDonalds and Honda.
Abstract : Such logos represents some abstract logo art form. Nike logo is a great example of abstract logos.