This can turn into a $1,000 question fairly quickly...
In general, the new partner will buy his/her interest from the other partner who is leaving. The partnership may close based on the ownership interest exchanged and become a new partnership all together, or stay the same ol' existing partnership.
The partnership may sometimes wish to make elections or special allocations under operating agreements between the partners, such as stepping up a basis on contributed property so that depreciation can be claimed equally. This can get tricky.
The partner who is leaving will recognize gain/loss on the sale of the partnership interest and the new partner will assume an outside basis equal to what was paid for the partnership interest. In the presence of an election, additional taxes may get paid.
Generally the partnership would have some sort of business valuation done. CPA's with valuation designations (ABV) charge big bucks for these, but they do protect the partners and help clarify fair value for tax purposes, especially for a related party transaction.
Let me know what else you need to know...
Thank you for your question.