I've done a lot of application reading over the years, although none in the last 3 years. This is a huge topic, but here a condensed (!) version of how I see it. The essay, letters, transcript, and GRE scores are the central parts of the application.
On the essay: I discount undergraduate math research (REU) experience, a topic often emphasized in the essay. I found no correlation between a student having participated in an REU program and their success in grad school. Dont assert that an REU gave you insight about what math research is, in 99.99% of the cases it didn't. Don't BS in your essay. "I am currently studying aspects of the Langlands program" is unlikely to be believable except in very unusual circumstances. Honesty and enthusiasm is what I valued most. What I want to read is "I love math and am willing to work like a slave for years to try to become a mathematician".
Letters: This is a very difficult issue, since letter writing is taken seriously by some writers and, are written by the student herself and signed by the professor in other cases. Honesty and specificity from the letter writer gives the letter more weight. Ask your professor to be clear and give specific comparisons to other students. The student should therefore make themselves familiar to their letter writers in the months before applications are due. If your professor says "Write a letter, I'll sign it" get someone else. Don't ask someone who hasn't had substantial mathematical discussions with you to write a letter.
Transcript: Good grades in all subjects indicate a serious student. Lots of math classes can be good, but is not always necessary. For students from the USA, strong performance in year long Algebra and pre-measure theory Real Analysis is the most important thing, followed by good performances in other theoretical classes. A few graduate level classes are a bonus, but not critical if there are plenty of serious math classes.
GRE scores: I valued a reasonable score (at my school, a research-1 University, reasonable means top 50%) on the advanced subject portion. Unless things have changed in the last 3 years, GRE scores from mainland China were not reliable, and skewed everyone else's scores down. Perhaps the problem has now been solved and the scores are more meaningful.