Both of these have the meaning of 'to give your word' to someone. 'Promise' is usually used in the context of an undertaking to do something in the future. For example 'I promise I won't let you down' or 'Promise me that you'll keep this a secret.' 'Swear' used in this sense is an assurance that what you are saying is true. In legal contexts, people are often required to swear on the bible or another holy book before signing important documents or giving evidence. In other situations, people might say 'I swear on my mother's life that..'. You can use 'swear' in the same sentences as the examples for 'promise' above ( e.g. I swear 'I won't tell anyone'). This is stronger than 'promise'. But you can also use it for situations in the present and the past e.g. 'I swear I don't know' or 'I swear I didn't see anything'. You wouldn't use 'promise' in these situations.