Could is usually followed by an infinitive without 'to': I'm glad you could come. Sometimes it is used without a following infinitive: I came as quickly as I could.
Could does not change its form, so the third person singular form does not end in '-s': She could play the violin when she was six.
Questions and negatives are formed without 'do': Could he help you? ♦ I could not breathe.
The negative form could not is often shortened in conversation or informal writing to couldn't: I couldn't find her phone number.
Could is often used in question tags: They couldn't see us, could they?
Could has no tenses, no participles, and no infinitive form.
There is no past tense, but could have followed by a past participle is used for referring to something in the past that was not real, or something that may possibly have been real: I could have been killed. ♦ What was that noise? Could it have been the wind?
Could is used as the past tense of can when it means that someone had the ability to do something, or that something was possible: The Roman army could march 30 miles in a day.
But if you want to say that someone actually succeeded in doing something, use was/were able to: Thanks to this grant, we were able to buy new equipment.