Optimistic locking.
Pessimistic is harder to implement and will give problems in a web environment. What action will release the lock, closing the browser? Leaving the session to time out? What about if they then do save their changes?
You don't specify which database you are using. MS SQL server has a timestamp datatype. It has nothing to do with time though. It is mearly a number that will get changed each time the row gets updated. You don't have to do anything to make sure it gets changed, you just need to check it. You can achive similar by using a date/time last modified as @KM suggests. But this means you have to remember to change it each time you update the row. If you use datetime you need to use a data type with sufficient precision to ensure that you can't end up with the value not changing when it should. For example, some one saves a row, then someone reads it, then another save happens but leaves the modified date/time unchanged. I would use timestamp unless there was a requirement to track last modified date on records.
To check it you can do as @KM suggests and include it in the update statement where clause. Or you can begin a transaction, check the timestamp, if all is well do the update, then commit the transaction, if not then return a failure code or error.
Holding transactions open (as suggested by @le dorfier) is similar to pessimistic locking, but the amount of data locked may be more than a row. Most RDBM's lock at the page level by default. You will also run into the same issues as with pessimistic locking.
You mention in your question that you are worried about conflicting updates. That is what the locking will prevent surely. Both optimistic or pessimistic will, when properly implemented prevent exactly that.