Groups are a way of organising keywords. You may want to organise keywords in a certain way to monitor your progress for different silos on a website, e.g. you might have two groups called "red widgets" and "blue widgets". You could also create groups for keywords that are actively having their search engine optimisation worked on so you can compare them to a control group of keywords.
Groups are also useful because they allow you to make notifications or reports apply to a subset of keywords and are useful for granting other users access to tracking and keyword data. Want to be notified when your keyword "the meaning of life" changes rank by more than 42 places? You can do it with groups.
A keyword can be in many groups and it will only count as a single keyword in your account usage.