At their simplest, contingencies are things that the "contract is contingent on." In other words, if the contingencies are not met, they are not "removed" by the buyer by a certain date. Should this occur one or perhaps both parties, may cancel the contract.
There are many potential contingencies in home sale contracts. First, a definition: close of escrow means that all the documents and monies required have been deposited with the escrow company (which is also usually the title company). Once that is done escrow if officially "closed" and the monies are distributed to the parties due (usually to the seller and to the seller's mortgage company) and the deed is recorded at the county office. Once this is done, the home is sold and the buyer is the new owner.
The seller has to be able to pass a clear title to the buyer (if they don't completely own it without a "cloud" on title, they can't sell it, so the contract is cancelled). The preliminary title report is reviewed by the buyer and if they find it acceptable the title contingency is removed. The seller must be able to pass clear title to the buyer for escrow to close.
HOME SALE CONTINGENCY
This contingency states that the buyer has a home to sell and it must close escrow prior to the buyer's home purchase can close escrow.
If a loan is to be used to purchase the home, there is usually a loan/mortgage/finance contingency. It simply says that if a loan is pursued in good faith by the buyer but cannot be obtained, the contract may be canceled.
The contingency states that if the home does not appraise for a specified amount, the contract may be canceled.
A buyer may elect to have an inspection done at buyer's expense. If the buyer finds items that are objectionable in the inspection, the contract may be canceled. Often, when legitimate problems are found, a Request for Repair is prepared and the buyer and seller agree to either have the seller fix the items, or offer a credit at closing so that the buyer may have them fixed after the close of escrow. A fine point here: quite often the seller has an inspection done as part of their disclosures. This is done in an effort to ensure that deficiencies in the condition of the property are brought out into the open and reduces the chances of litigation later.
SELLER PURCHASE OF A HOME
This is a seller's contingency that states that the seller must find a new home before the buyer's purchase of the seller's old home may close.
SELLER'S MANDATED AND CONTRACTUAL DISCLOSURES
The seller has a variety of mandated disclosures by law as well as some disclosures that are obligatory as a matter of the sale. All of these materials must be submitted to the buyer and adequate time given for review of same.
This is not really a contingency that must be "removed" by the buyer prior to close, but rather a condition. The seller must transfer the property to the buyer in the same general condition as when the buyer went into contract allowing for any repairs that have been agreed to. Interestingly. this does NOT include patching of holes created by hanging items in the home.
WAIVING OF CONTINGENCIES
Buyers often waive some of the contractual contingencies in an effort to make their offer appear better and less hassle to the seller. Sometimes the inspection contingency is waived (particularly if the seller has already had an inspection done) although that presents some danger, and every brokerage insists that the buyer sign something that says in doing so they are going against the broker's advice. The appraisal contingency is sometimes waived on all cash deals or very high down payment purchases where the buyer is convinced the property is worth at least what they are paying for it (in the very least worth it to them, anyway). The loan contingency is sometimes waived by highly qualified buyers with high down payments.
All waivers of contingencies pose some risk to the buyer - be sure to have this discussion in depth with your agent so you fully understand those risks.