I think the question you're asking is actually really complicated since it depends what the verb is "doing" in the sentence.
But usually "ta" is the informal, past tense ending of ichidan verbs (verbs ending in -eru or -iru, unless they're godan exceptions like kaeru/ 帰る). "Ea" as an ending appears as "nda" at the end of godan verbs ending in -bu or -mu in their present tense, informal dictionary form [e.g. asobu (to play) or amu (to knit)....would be asonda (I played) or anda (I knitted) ].
Additionally, "da" is the informal present tense of "desu," roughly meaning "is". So, I could say "watashi wa amerikajin da" meaning, I am an American.
So, "a" endings are generally past tense.
Te or "de" forms on the other hand are often used when the verb is part of a larger grammatical structure. For example, if you wanted to say, "I am playing" (present progressive, as in you are at this moment playing and continuing to play), you would say asonde iru/ あそんでいる. Or with an ichidan verb such as taberu, you would say tabete iru/食べている--I am eating.
The te/de and ta/da are not used with basic negatives. To say that you will not play, you would say asobinai. To say that you will not eat would be tabenai. You could say "I am not eating now" by saying ima wa tabete inai, but that's the present progressive and the negative signal is actually in the nai part.
Da and ta are not formal endings. The masu endings are, e.g. tabemasu/asobimasu (present tense).
Whoo, I probably bored you do death. Granted, all this is from a Japanese learner, so I provide with it my caveats. =)