Register often refers to the degree of formality of language, but in a more general sense it means the language used by a group of people who share similar work or interests, such as doctors or lawyers.
For formal and informal register; 'Would you mind passing the salt?' is appropriate for a formal situation with strangers, whereas 'Pass me the salt' would be used for a situation where friends are talking, or possibly when being rude.
In the classroom
Although register is a complex area, the most common aspect looked at in language learning is the degree of formality. Teachers often divide functional language into three working categories, formal, neutral and informal.