It appears (comparing entries at http://en.wiktionary.org/wiki/Unterricht with http://en.wiktionary.org/wiki/Klasse) that Klasse is very much synonymous with "class", conveying the sense of a group, whereas Unterricht seems to be an individual lesson or single instance (e.g., "today's class").
NB: I am basing this solely on those two entries at the single source, wiktionary.org.
It's still not clear. In some languages there can be at least three different words that could all be translated to "class" in English.
I can sign up for a class, which would be a specific course and a set of corresponding sessions.
I can be in a class, such as the class of 1997, or the class could refer to all the students who study a particular subject at a given level, who meet at the same times and places.
And a class can refer to an individual session.
It would be fine if they gave sentences such as "he was in my biology class," or "I went to class today," or "I signed up for a history class." But it would also have to be clear when it would not apply.