I was leery to post, but since I referenced Amdur Sensei's article in another post, I thought I should.
As best as I can tell, irrimi is one of those screwy things in aikido where we have a physical technique as well as a philosophical principle. We learn one to understand the other and I think there are already some good posts on this thread about that, including the article itself.
My understanding of irrimi is similar to Fred's definition. In this sense, "irrimi" is the concept of entering into our opponent's body in order to connect and control her center. Physically, this concept can be executed in a number of ways. In any case, the spirit of irrimi is to confront the attack not evade it. The technical achievement of irrimi is to displace the attack and dissolve its power.
I have included a Friendship demo of Sunadomari Sensei. He was a big fan of this "greeting" irrimi and you can see the entering movement in many of his techniques. You'll also notice he spends time talking about removing power:
We talk about marubashi, the concept of your partner abandoning the line of attack while you maintain a direct line [of attack] into your partner. Amdur Sensei talks about occupying space and that is a key concept of irrimi for me (I am trying to get Ledyard sensei to write a piece on this topic). I don't care where you put your feet, if you don't enter into your partner's space and take their center you are not doing irrimi.