Received a PM asking me to reply.

The best way to discuss these kinds of issues is by posting specific problems that you have seen. It's very difficult to write a grammar lesson about an entire topic in a forum post - that could take an hour. I will touch on some topics in this post, but I can't construct an entire grammar lesson here - if you have further concerns about specific problems, please post those problems.

First, the "er" ending of both words is a tip-off that we typically use these words in comparison situations.

The word "lesser" can be an adjective or an adverb - that is, it can be used to modify a noun and it can also be used to modify other parts of speech. In a comparison situation, one thing is "lesser" somehow than another. For example, there are two evils. One is the greater; one is the lesser. It can also be used to discuss comparative rank - the "lesser" man deferred to the more experienced man. It is probably most typically used in certain idiomatic expressions (see kmittal's first and third examples). Many times, you are often able to substitute "greater" and still have the phrase make sense (the greater of two evils, to a greater degree). The fact that we talk about the greater and lesser of two evils, rather than the higher and lower of two evils, is just an idiomatic usage issue. Finally, greater and lesser will more typically be used with uncountable nouns.

I'm trying to think of a specific OG or GMATPrep question in which I have seen this word tested, but I can't think of one. Can anyone else?

The word "lower" is most often used as an adjective or verb. In adjective form, it is again typically used in a comparison situation. Something is lower than something else or something is lower than it used to be. The table is lower than it used to be because I chopped off the legs. Her score is lower than his. The lower bookshelf contains the book you want, not the higher bookshelf. For this one, you can often substitute "higher" and still have the phrase make sense. Higher and lower are more typically used with numerical comparisons.