The first thing is, look them up individually. www.thefreedictionary.com gives example phrases.
If you're still confused, remove the -ment part and look up each word again. You'll quickly see that "attain" is very different from "accomplish", for example.
I'm not sure how to answer your question about which one to use. Could you explain it using different words? I'm actually not sure who makes the request, for a start.