Puedas jugar?
It means, "the chess." Frequently, Spanish uses definite articles in ordinary speech which are not translated into English. Example: ¿Te gusta jugar al [a+el] ajedrez? Do you like to play chess?
Playing tennis in Spanish is jugar al tenis.
"ajedrez" is "chess" in English.
There is no difference between: Jugar al beisból and jugar el beisból. It depends on the country, but both forms are used.
Do you prefer to play baseball or football? Para mi, prefiero jugar al beisbol.
To play tennis.
"Can I play basketball?"
jugar al baloncesto
To play the white. Possibly in chess?
As a statement "you like to play tennis", it would be "Le gusta jugar al tenis." As a question "do you like to play tennis?", ¿Le gusta jugar al tenis?" Spoken, the differrence is just in inflection - with the voice rising at the end. If you are asking someone if they want to play tennis, the it is "¿Le gustaría (Or "Quisiera") jugar al tenis conmigo?"
In Spanish, you would say, "¿Sabes jugar al ajedrez?" In French, it's "Sais-tu jouer aux échecs?" In German, you would say, "Weißt du, wie man Schach spielt?" Each phrase translates directly to "Do you know how to play chess?" in their respective languages.