Redundancy is using more words than are necessary. A redundant sentence will still mean the same thing if the extra word or words are removed. The definition from Merriam-Webster is "an act or instance of needless repetition."
Redundancy in English usually happens when an adjective is added to a verb or noun that means the same thing. A speaker or writer is often trying to add emphasis or attempting to thoroughly explain something when they make the common mistake. In general speaking a redundancy might be overlooked, but it could cause you problems in a school paper or business letter.
Less is More
A benefit to reducing redundancies in writing is that it helps make your writing more clear and concise. Sentences that are wordy take longer to read. Looking for redundant phrases in your paper or article is an important part of the editing process. Don't slow down your reader. Help them quickly absorb the information you are providing.
Pleonasms: Redundant Phrases
An obvious example: If you said something was "obviously obvious" that would be redundant. No one (we hope) would ever use this particular phrase unless they were being funny, but this blatant error gives you an idea of how redundancies occur. Some redundancies that are nearly as ridiculous are used frequently in English. Pleonasm is word that means a redundant phrase. An example the Dictionary.com definition of pleonasm gives is "true fact." A fact is true by its nature so the addition of the "true" adjective is not necessary.
The Purdue Owl has an article about eliminating unnecessary words to make your writing more clear. It includes tips for finding redundant pairs and unnecessary determiners and modifiers. Some examples of redundant pairs the Owl provides include "basic fundamentals, end result, free gift" and "sudden crisis."
Comedian George Carlin often made fun of silly expressions during his comedy routines. Language Trainers has compiled a list of some of the redundant phrases he provided in his book Braindroppings. They include "total abstinence, advance warning" and "lag behind."
Here's a video from Grammar Gripes spreading awareness about redundant phrases, including "future plans, usually always" and "in the process of."
Lists of Redundancies