The first thing that comes to mind when failing to follow someone on Instagram is that you’ve been blocked by the one you tried to follow. If that is the case, try asking the person to unblock you, good luck with that, treat them to something nice if you are really desperate, might be worth the effort.
On some occasion, however, the application might encounter a problem in the system, usually after you followed and unfollowed a mass of people in a short period of time. Instagram will prevent your account from doing so for a few days, since it thinks you are a “bot” designed to do so, the same kind that allows anyone to buy a service for tons of instant followers and “likes” on their account. Just wait for a few days and Instagram will sort it out, and let you follow anyone again. It’s a measure to prevent their servers from clogging up. So don’t overdo it. You can’t follow more than 7500 people too, but if that’s the case, a message will show up that says “You can’t follow any more people”. Or try to restart your phone. That always helps on any electronic devices. If all else fail, it’s probably a temporary error that will be fixed sooner or later.