Leave is interchangeable with let when followed by alone with the sense "to refrain from annoying or interfering with'': Leave (or Let) her alone and she will solve the problem easily.
If you leave someone to do something, you go away from them so that they do it on their own. If you leave someone to himself or herself, you go away from them and allow them to be alone. I'll leave you to get to know each other. Diana took the hint and left them to it.
To leave is to go away or exit. You can leave the country, the house, or a job. This has many meanings, most having to do with departing. Getting permission to take time off from work is taking a leave of absence. To not disturb something is to leave it alone. To abandon or forget something is to leave it behind.