How can I update my games ? (Title update / patch)

When you are connected to xbox live, it checks for updates every time you start a game. If there is one, it prompts you to download it before you can play. You can set up the console to skip asking you and automatically install any necessary updates too.

Also, if you buy a game digitally, they already come fully updated.

Xbox 360 games generally have very small updates. The 4mb update for GTA IV is correct. When you see the message regarding a game update when connected to Xbox LIVE, this will be the latest update for that game, regardless of the download size.

Title updates on the 360 were limited to 4MB because there were systems that had no storage other than a tiny memory unit. The Xbox 360 also supported delta patching, so if you were patching a large file, only the differences in code between the old file
and the new file needed to be included in the title update, while on the PS3 you would need to download the entire new file even if only 1 line of code was changed, which is why patches were way, way bigger on the PS3. From what I've read, the patching system
on the Xbox One works like the PS3's, although I don't know if that's true.

The 360 way of doing things was good in some respects - it was great for users, because 4MB title updates download in seconds, even with slow connections, and devs knew that they wouldn't be able to use massive patches to patch horribly broken games that
were released too early. The downside was that the title update sizes limited developers, especially as they were only allowed to release so many updates for a game (I think it was 2 or 3) and they had to pay for each one to go through cert. Also, you obviously
can't add in considerable new content to a game such as new maps, modes or characters via a 4MB title update. For instance, Killer Instinct on the Xbox One got an update that boosted the resolution to 900p, which you wouldn't be able to do with a tiny title
update. Some exceptions were made to the 360 patching rules later in the console's life cycle, so games got title updates bigger than 4MB and some games got half a dozen or more title updates, but it was usually big games like COD that those exceptions were
made for.

I don't know how patching works on the PS4, but it's had its fair share of giant patches too. The Sniper Elite III issue was explained by the devs and wasn't down to the difference in how the Xbox One and PS4 do patching.

It might be background download. You said update, I assume it was a title update(patch) for a game. I think it didn't finish because updates don't work in background download mode.