actually you can also omit "the" in the first sentece. you use "the" when you refer to a specific thing. for example, in the first sentence, the speaker refers to a specific guitar, because maybe there are many types of guitar. But if you want to ask for "playing guitar" in general, you just need to omit the "the".

Further to what Alex Pugh has said [above],  you can't say  'Can you play the tennis?', or use 'the' when talking about any kind of game.

However, when referring to a musical instrument, you can either use 'the' or not, although I think it sounds better with 'the'.

If you ask someone 'Can you dance . . . ?, you should use 'the'  e.g. 'Can you dance the samba ?'

You talk about 'listening to the radio' but 'watching television', although in this case, they are reversible :  you could say 'listen to radio', or 'watch the television' although they don't sound quite right to me.  As Alex has said, there is no grammatical reason, and it comes down to what feels right.

You can say "He plays the guitar,"
you <em>can</em> say "He plays guitar,"
yes, you can <em>also</em> say "He plays a guitar,"
and you can say "He's a guitar-player"
and "He's a guitarist."

Languages are like that. When I was in high school I was angry at Spanish for having irregular verbs. What can you say? Languages are what they are.

This may sound useless, but there isn't really a grammatical reason. Sometimes, English is just random. Phrases with and without the 'the' are interchangeable. After exposing yourself to native English conversation, you will gras the idea and just use whatever is most natural within context.

Thanks everyone for contribution to my disscussion. Your answers one more time has convinced me that the articles are the most complicated part of English.

I love your answer Adrian - feel right for you or for me...  :D