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There are many different ways, including transaction queueing, optimistic concurrency control etc. This is actually a very complex question, there are books written about it:
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It depends on the level of logging in the database. If strict write-ahead logs are kept then in the case of a system crash, the database can be wound back to a consistent state.
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It depends on the type of concurrency. Optimistic concurrency involves no locks, but if the state of the db has changed once the transaction has finished, it is abandoned and restarted. This can speed up dbs where collisions are rare. There are also different levels of locking: row,table, or even the entire db.
These are very complex questions, I’d advise buying a book, or attending a concurrent systems lecture series if you want to be able to fully answer them 🙂