"I wrote an application."
Translation:Jag skrev en ansökning.
None, they're synonyms. But ansökan cant be pluralized, you'll have to say ansökningar if you want to refer to more than one.
Is this "application" in the sense of, e.g., an application for a job or a grant, or as in a program, or can it be used both ways as in English?
Those are different things in Swedish, but since we don't know what is meant without context, several different answers are accepted. en ansökning is an application for a job or grant; 'a program' is ett program or sometimes en applikation, 'an application' in a more general sense ('the process of applying something' or 'the use of something') is en tillämpning (sometimes also used about computer programs).
How come skrev doesn't have a de or te ending? Is it just the way the language is and u have to memorize all verbs different tenses or is there a general rule for determining how a verb changes?
Some verbs form past tense through a vowel change, just like in English (compare write -> wrote). Those verbs have to be memorized.
Here is a good list of strong and/or irregular verbs: https://en.wikibooks.org/wiki/Swedish/List_of_strong_and_irregular_verbs