So you're at an adoption. What now? I don't know what other people look for, but here's what I do when going to adoptions:
This big pinto was in a pen with a bully horse. The bully would seek out the pinto, demanding a fight. When pushed, the pinto would give in and the two would rear, strike out, and reach out to bite each other, just like stallions in the wild. But the pinto would have preferred to be left alone. Other horses followed him, which irritated the bully. Sound familiar? Kinda like kids on the playground. The bully horse was jealous of the natural leadership of the pinto; he wanted to lay claim to the 'herd' that was following the pinto around the pen. The pinto was a smart horse. He watched and knew where the bully was at all times, doing his best to stay out of the way. I found this to be a very positive thing with this guy.
Some horses get curious about the people who come to see them. They'll come up and sniff and eventually let you rub them through the corral rails. I have mixed feelings about these horses. I like a horse who may sniff, but uses caution over a horse that suddenly falls in love with the crowd of wither scratchers, and this is why; the cautious horse is going to give you more space once home and training begins. The overly friendly horse may become pushy; refusing to give you space. Not that it's always like that...it's just my personal preference when selecting between the two. The younger the horse, the less trouble it will likely pose; but if it's an older horse, I would personally select another. The filly pictured above finally got brave enough to reach out for some hay once all the other horses in her pen had been taken home. Without any other companionship, she turned to a friendly human.
The more time you're able to spend watching the horses prior to the adoption, the better idea you'll have about which one you may want to go home with. There are always plenty of people available to ask questions to. Both BLM employees, volunteers, and past adopters attend these events and have plenty of stories and advice to share. Listen to everyone...but in the end, you'll need to select the horse that speaks to your heart.