Like its English counterparts "can" and "could," poder functions as an auxiliary verb, although in Spanish it is followed by an infinitive. With a few exceptions, most of which don't have counterparts in English, it can't stand alone.
Here are the various ways poder is used:
In the present tense to mean "can" or "may": The present-tense forms of poder indicate the ability, either physical ability or having permission, to do something. It is distinguished from saber, which means "to know how." Thus, while one may ask, ¿Puedes tocar el piano hoy? ("Can you play the piano today?"), one normally would ask, ¿Sabes tocar el piano? ("Can you play the piano?" or "Do you know how to play the piano?").
Examples: Puedo hacer lo que quiero. ("I can do what I want.") No puede trabajar los domingos. ("She can't work on Sundays.") No puedo ir al cine. ("I can't go to the movies.")
In the future tense to mean "will be able": This is similar in usage to the present tense.
Examples: Podré hacer lo que quiero. ("I'll be able to do what I want.") No podrá trabajar los domingos. ("She won't be able to work on Sundays.") No podré ir al cine. ("I won't be able to go to the movies.")
In the preterite or imperfect to mean "could" or "was able": Which tense you use depends on whether the reference is to a one-time event (preterite) or something occurring over a period of time (imperfect). In the preterite, poder can have the sense of "to manage to."
Examples: Pudo salir. ("He managed to leave.") No podía salir. ("He was unable to leave.") No pudo trabajar porque dormía. ("She couldn't work (that particular time) because she was sleeping.") No podía trabajar porque dormía con frecuencia. ("She couldn't work because she was often sleeping.")
To make polite requests: As in English, such requests are made in the form of a question. Usually the conditional form of poder is used, but (while it may seem illogical) the imperfect also can be used.
Examples: ¿Podrías darme un lápiz? ("Could you give me a pencil?") ¿Podías darme un lápiz? ("Could you give me a pencil?") ¿Podría lavarme usted los platos? ("Could you wash the dishes for me?") ¿Podía lavarme usted los platos? ("Could you wash the dishes for me?")
To express possibility or suggestions: Either "could," "may" or "might" can be used to translate poder when it is used to indicate a possibility or offer a suggestion. In such cases, either the conditional form of poder or (again, seemingly illogically) the imperfect can be used. The imperfect form may be understood as more colloquial.
Examples: Podríamos ir al cine. ("We could go to the movies.") Podíamos ir al cine. ("We could go to the movies.") Podía no haber salido. ("He might not have left.") Podría no haber salido. ("He might not have left.")
To espress what could have happened but didn't: The preterite is usually used in such cases, although the conditional can be used when directly criticizing somebody.
Examples: Pudo salir a las tres. ("She could have left at 3 o'clock.) Pienso en lo que pudo ser. ("I'm thinking about what might have been.") Me lo podías haber dicho. ("You could have told me.")
Keep in mind that poder is irregular. The o in the stem changes to u or ue when stressed, and the ending is shortened in the future and conditional tenses.