'Check on' and 'check up on' are slightly different in meaning, with the latter denoting a situation requiring more attention. For example, a mother would check on a sleeping baby but check up on a sick baby. Both phrases loosely translate to 'make sure ___ is alright' (one to a slightly greater degree than the other) and are used to describe an object or situation that does not presently hold your attention. For example, I could not currently say I'm going to 'check on/check up on' my computer because, at this moment, I am using it.
Both phrases are often preceded by 'I'd better'. So...
I'd better check on that soup I'm cooking.
(I'm going to have a quick look at that soup I'm cooking.)
I'd better check up on that soup I'm cooking.
(I'm going to have a quick look at that soup and I anticipate it will need some attention.)