All the dictionaries already mentioned are great. Those looking to add a slightly different approach to their arsenal may be interested in the following. One thing I own is a hard copy of An Foclóir Beag (Gaeilge – Gaeilge). I wouldn’t recommend this as your primary dictionary, as it obviously doesn’t give any translations of the words. What it does do is give you a brief definition or, in some cases, synonyms and phrases using different senses of the word in Irish. If you’ve used the Foclóir Beag site, it is basically the exact same content without all the declensions and conjugations spelled out for you (there is a reference for the irregular verbs, and the gender and declension category for each word is given, obviously).
The value of this is that I think it helps draw associations in your brain between different Irish words, essentially getting you closer to thinking in Irish. It is a cheap way of approximating some aspect of the experience of communicating with a fluent speaker who doesn’t speak English (obviously something you probably won’t have to really deal with in Irish). In other words, if you had to ask what “madra” meant and got the answer in Irish, that is a more powerful mental exercise than simply being told “dog”. Being told “ainmhí a choinnítear chun aoireachta nó seilge nó cosanta, nó mar pheata” makes you learn not only what the word symbolizes but also how it relates to other words and allows you to build a more complete picture of the most frequently used words in the language, i.e. those that are necessary to talk about most things.
I would actually encourage people to try as much as possible to look up a word first on Foclóir Beag and see if you can figure out what it means before you look up the translation. I think you’ll find it rewarding and you can make kind of a game out of it. If you don’t know a word in the definition, look that up too. You’ll sometimes find yourself going off on a tangent looking up 10 words just to figure out the meaning of your original word. But now you’ve associated 10 words with one another instead of learning them randomly on their own. I only wish they would publish a “Foclóir Mór”, with extensive definitions for each word in Irish.