A contingent offer simply means the offer is "contingent" on selling your current house. So, the seller has to be willing to wait for you to get a contract on your house before you can close.
The underwriting would be no different then any time you are selling and buying at the same time contingent or not. The underwriter will just assume you have sold your house eliminated the current housing current payment from your debt to income ratio and assuming the net equity for the down payment. Of course, you have to prove your house did sell before you can close on the new house but that can all be done the same day.
As you might expect contingent offers are not as popular with sellers then non-contingent offers and it just depends on the house, the sellers and most importantly the local market if a contingent offer would be accepted. It never hurts to try if that is your best option.
But, you might also want to explore other options as a plan B just in case. One of those options would be a bridge loan to eliminate the need for a contingent offer in a competitive market. More info here:
residentialbridgeloan.com/