Business partners aren't like employees. They can't just up and walk away from a properly run business. This is because they have typically invested capital into the company, both in terms of money, property and "sweat equity" over the years. These contributions need to be accounted for when a partner, shareholder or member leaves the company. There also may be legal formalities you must go through.
Specific procedures vary by state, but you may have to file amended articles of incorporation, amended articles of organization or an amended annual report with your state's secretary of state's office. There is a nominal fee to file these amendment forms. You will need your file number and business name, along with the names and Social Security numbers of all remaining principals.
If you are running a partnership, corporation or limited liability company, your partner legally owns a share of the company. Even if your partner quit working tomorrow, you would still have to pay that partner her share of any dividends you pull out from the company. To avoid this, you have to buy her out. If you have been doing proper accounting, the partner will have a "capital account" with the company, which accounts for contributions of money, property and time and represents the total equity a partner has in the company. Many businesses have an operating agreement that details how the partners will agree on a fair price for the business. This could be a multiple of earnings or revenues, agreed to in advance.
If possible, you will also want to get a statement in writing from the departing partner that she considers the buyout to be a settlement in full of any outstanding claims against the company. This can go a long way in preventing any lawsuits down the road. You will also need to address any potential claims against the company for which the departing shareholder may have personal liability.
Corporation owners should keep good records of the meeting at which the shares of the company changed hands. You must document the transaction, including the number of shares, the price and the date. Get a letter from the departing partner resigning from all corporate offices held.