One of my users let the cat out of the bag and told me they were using one of my free apps, which is monetized by ads, but they were blocking the ads with an ad blocker. The
I'm sure this answer won't be entirely popular with certain segments of developers, however consider if you fall into this category that perhaps your app doesn't deserve to exist on the app store. Please note that these are all implementable as code changes, no hackery or spyware like behavior required.
Basically, change the economics of your app. The User is Always Right - this is the attitude taken by one of the most successful advertising companies ever (Google). If your ads are being blocked by users, its because you suck, not because ads or ad-blockers suck.
http://books.google.com/books/about/The_User_is_Always_Right.html?id=gLjPMUjVvs0C
I re-iterate: all of the above changes are things you can legitimately do in code to prevent anti-ad behaviors. Ads are blocked for security reasons and visceral reactions, primarily, and sometimes bandwidth and performance, so make sure your ads don't invoke any of these problems, at the code level.
Finally I did want to touch on what Borealid said, which I re-iterated above; in the end it is a 'cat and mouse' game, because the user has ultimate authority and responsibility, both legally and morally, over their own property. A user can do whatever, including directly modify code on the fly. Of course, there are restrictions you can implement etc. but there are always ways to get around the problem. This is the ultimate problem (technically) with DRM (which is what you're trying to do). Rather than waste time and effort on this game, it is better to encourage users to keep ads around; they'll become your best, smartest anti-ad-blockers, for free.
This is an extension of a previous answer. The user has informed me that the app they are using is called AdFree Android. It can be found on the market. The app says it works by "nullifying requests to known hostnames serving ads."
I suggest that if you monetize any of your apps with ads, you check at startup for this program and give the user a nasty message, then terminate your app.
The top two answers help you with only a particular (if, probably, the most popular) method of blocking ads. Root users can also block ads with a firewall on the device. WiFi users can block ads with an upstream firewall.
I suggest:
Don't reward ad-blocking users. Ensure that your layout reserves part of the display for an ad even if one can't be loaded. Or if you have a full-screen ad that plays for a bit, ensure that your app waits for a bit even if the ad can't be played. If you use notifications as adverts (you scum), notify the user when you fail to get such an advert. This could be read as "annoy all of your users", but your normal users know what they're getting, and your ad-blocking 'users' aren't wanted.
Ask ad-blockers to stop. The less proftable an industry that supplies what a user wants, the less that industry will supply what the user had wanted. An individual developer will find that he makes more money serving other users. You know this, and your users will think it obvious after you tell them, but it's still an economic argument - it's not intuitive. Have a backup ad that says something like, "This is my job. If you don't pay me, I'll get another one, and you won't get more apps like this from me."