They are playing on the mobile phone. --> This means there is only one phone and all of the students are sitting close together playing with just one phone. They are playing on their mobile phones. --> This means that each student has their own phone and is playing with it.

It's correct to say mobile phones, with the plural S, when you talk about "they", more than one student. The students can be playing on the bus, as they are sitting "on" the bus, but the object they use is "with". "They are playing with their mobile phones". If you say "the mobile phone", it suggests there is only one phone, and it is a special phone. "The" is a definite article. while "their" is a possessive pronoun. "Their" refers to who owns the object / objects.