Assessments are the card networks' own cut of every transaction. They're separate from interchange (which goes to the cardholder's bank) and from your processor's markup. Visa charges around 0.14% + 2¢, Mastercard 0.1375% + 2¢, Discover 0.13% + 2¢. Like interchange, every processor pays the same assessment to the networks — there's no negotiating it.
What to watch for: some processors lump assessments into their markup line on tiered pricing statements, making it look like the processor is charging more than they really are. Others itemize them. On interchange-plus pricing, assessments are usually shown as a separate line.