An omnibus bill is different from an ordinary bill because an ordinary bill is about one topic. An ordinary bill might be, for example, about whether to implement background checks on people who want to buy guns. By contrast, an omnibus bill is about a large number of unrelated topics. It might include the provision about background checks, but also have money to repair a bridge somewhere, language about immigration reform, and other such things that are not really related to one another. There is no precise line between an ordinary bill and an omnibus bill.
Omnibus bills are created out of political necessity and out of time pressures. They are typically thrown together so that Congress can try to get a lot done before its session ends. However, they are also made for political reasons. It is much easier to hide a measure in an omnibus bill and pass it because no one has read the whole bill than to get the measure passed on its own. So, for example, the money for the bridge mentioned in the previous paragraph might be a lot easier to pass if it is tucked into a huge bill than if it is on its own.
Please read this link for an excellent discussion of omnibus bills and the logic behind them.