You need silver paws to buy new guardians. These can be obtained by taking part in game events, and they also drop in event locations.

You can see how many silver paws you have in the Inventory and in the guardians window.

When you have the required number of items for a purchase, press the guardian on your desk, select the guardian you want from the general list, press the purchase button (the guardian's price is displayed on the button) and the guardian will immediately be yours.