This is more common than ever because of poor quality materials. A washing machine door hinge should never break, and for decades they never did, but they do now. The washing machine door and hinges are both easy enough to replace so it should be mostly just a question of buying a new one and replacing it.
Replacing a door hinge
A washing machine door hinge is usually held onto the casing with two screws, bolts or even rivets. They can be large self tappers, or bolts, with nuts or plates behind them which can drop inside the machine when unscrewed. Sometimes the hinge will hang on the casing with lugs, or hooks which slot into holes and the screws are just extra securing.
So the main thing to watch out for is unscrewing the hinge only to hear something drop inside, which is awkward to retrieve. Many washers have base plates fitted, or have a chassis where parts can fall and be difficult to retrieve. Before removing the door hinge, disconnect the washer from the mains and remove the lid. If you can see the back of the hinge and reach it, undo the screws and catch the fixings or plate they screw into.
If it’s difficult, try peeling the door seal away so you can get very easy access to the back of the hinge. The door seal is usually secured to the shell at the front with a metal or plastic band. It shouldn’t be too difficult to remove and refit on most washing machines.
If you can’t see any fixings at the front you’ll have to see if you can locate anything at the back, bearing in mind the advice above.
Something has dropped inside the machine
In the last paragraph I advised to beware of unscrewing the door hinge without checking if it has a loose plate at the back that will fall inside the machine. If this has happened to you the part that has fallen inside will most likely be stuck in a chassis. To retrieve it you need to make sure the appliance is disconnected from the mains (read on..)
Here are the various ways to retrieve the hinge part
Pull the washer forward and tilt it back to have a look underneath. You never know it could have just fallen onto the floor (relatively unlikely though). Pull the washing machine forward and tilt it back against a worktop or get someone you trust to hold it. Try to reach under and inside to find it. Warning, washing machines are very heavy! Do not let one fall over!
Remove the retaining clip or spring holding the door seal onto the door frame and peel back the door seal. Try to reach in and down. Disconnect the washing machine’s hoses. Pull the washer right out and lay it down on its front to gain better access to underneath
As a last resort, remove the door seal and kick panel strip, remove screws and take off the front panel. (There may or may not be more steps to taking a front panel off depending on make and model)