Over the weekend
Over the weekend means the same thing as during the weekend. It is used to talk about something that happened (or will happen) between Friday evening and Sunday evening.
Below are some examples showing how native English speakers typically use over the weekend. As you can see, the thing that happens during the weekend can be something quick, like a phone call, or something that lasts most of the weekend, like a trip to Miami.
On the weekend
The meaning of on the weekend is similar to the meaning of over the weekend, although it is not used quite as often. The two expressions are usually interchangeable and choosing one or the other is mostly a matter of personal preference. In all of the examples below with on the weekend, over the weekend would also be correct.
On the weekends (plural)
One clear difference between over the weekend and on the weekend is that only on is used with the plural: weekends. For example, these are all good sentences in English:
while these sentences would sound very odd to a native speaker:
At the weekend
At the weekend is a British English expression, which is used the same way as on the weekend in American English. Speakers of American English may understand this expression, but they do not use it.