On a cellular level we don't have the capacity to 'think' in the same way me and you would understand it. Our white cells would need a brain for that. It's more of a fundamental chemical thing, one cell releases certain chemicals and another responds with because the chemicals have stimulated a particular receptor that signals the cell to do it.

I'l also explain what /u/croutonicus said in much simpler terms, seen as you said you didn't understand it - it's been a while since I've had to crack out my immunology, but I'll give it a whirl. If I get anything wrong, please people correct me!

So helper T-cells are like scouts. They each have been given a 'picture' of one specific enemy who they can recognise if they get near them. This 'picture' is that capability to recognise a specific protein (antigen) on the enemy's cell surface. We have millions of these scouts running about, but they each only recognise one enemy each, and they don't really have any effective weapons to kill them. Esinophils are like infantry. They can't recognise an enemy, but they do have a radio and lots of weapons. If an enemy walked past the infantry, they wouldn't attack unless the scout flagged them up. Also if an enemy walked past a scout who didn't have his 'picture', nothing would happen either. When a scout (helper T) sees his specific enemy who's picture he does have, he radios to the infantry (via chemical messengers) the location of the enemy, and the infantry come in and attack.

The scout does has a secret weapon however; he can modify the battleground to suit his and his infantry's needs. He does this by making the roads (blood vessels) bigger so his army can flood in better and attracting more resources (blood) for his army. This is inflammation. This is the reason an inflamed area looks red, the blood vessels are dilated and lots of blood is rushing there.

There are also lots of other special 'units' that the scout can call in (such as B-cells and killer T-cells), and also lots of other 'scout' type cells (such as neutrophils). The specific immune response really depends on the type of infection, but because these were the only ones in the original explanation I'll leave it at that.

A really good video explaining the basics of the immune system is by Kurzgesagt. In fact their whole channel is just fantastic, I'd recommend watching it all!