Pending deposits are either physical payments from members for which payment hasn't been received yet, or credit card payments that have only been authorized and are awaiting your approval. For example, if a member selects to send payment through PayPal, but doesn't actually submit the funds through PayPal, the payment will be a pending deposit.
You can manually process all pending deposits through the Transaction->Pending Deposits menu, which will look like:

This table shows all list of all pending deposits in the database that are waiting to be processed. Approving a pending deposit through this menu is the exact same as if a member physically submits the payment. You can process pending deposits by simply checking the desired transactions, and completing the few simple fields on the HTML form.
The "Process Through Merchant Account" field is only applicable if you're using a real-time credit card processor, and specifies whether or not to send a request to the merchant account to capture the funds on the credit card. Generally, this option should always be set to yes. This will automatically send a request to the merchant account behind the scenes, the capture the previous authorization and actually charge the credit card.