If you allude to something, it means you refer to it indirectly, without saying it specifically. For example, if there is a software program with lots of bugs and errors, the developer might allude to the problems by saying "The process of developing the software has been very challenging." - He does not mention the problems, but he hints at them by describing the development as "challenging."

If someone has just received a promotion, they might allude to it by saying they're happy about recent events at work. Again, the person didn't specifically say "I'm happy because I got a promotion," they just referred to it very indirectly.

The word elude means to avoid or escape from something - we often talk about criminals eluding the police or eluding capture.

We can also use elude in a mental sense, when something "escapes" your understanding or cannot be gotten by you:

I've tried and failed many times; success seems to elude me.

Despite hours of study, the answer to the question continued to elude the student.

In fast spoken English, allude and elude are often pronounced the same.

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Hillary Clinton Secretary of State

Playing a critical role on the international relations scene, Hillary has managed to strengthen American alliances in Asia and Europe while simultaneously engaging emerging powers like Brazil, India, and China. Moreover, she has helped direct an administration that is surprisingly united on foreign policy.