In Game of Thrones, who had the right to claim the throne after Robert's death?

There's no simple answer to this question. Generally, the succession follows from father to eldest son. In this case that would be Joffrey, but since Joffrey wasn't biologically Robert's son (he was Jaime's), technically he wasn't the rightful air. Therefore, Stannis, as Robert's eldest living sibling, had a legitimate claim. However, Robert had plenty of bastard sons (such as Gendry) who also could've potentially had a claim, if they were able to be legitimised. But the likelihood of that was slim since those young men would likely not know their true parentage, have no knowledge of politics or what it takes to rule and probably wouldn't be very favourable amongst the people or the key players in Kings Landing. So overall, I suppose Stannis comes out as having the most rightful claim.

On Robert's death Stannis is the lawful King. Joffrey is not the son of Robert but of Jaime Lannister and so has no claim to the throne.

The line of rule usually descends from father to eldest son, but when that son is not of a suitable age to rule, its not unusual for a regent to be appointed, and in a patriarchal society, its far more common for a male relative of the deceased king to be assigned that role than the mother of the prince. Of course, power being what it is, its not unusual for anyone who feels they have a claim to want to seize control of the crown, and any male of the same line as the current or former king would feel they had a very strong claim, so its fairly common practice for there to be a lot of infighting with any dynasty.