Fear

"Fear. We face it every day of our lives. Sometimes, fear can be a friend who fights by your side, making your reflexes quicker, senses sharper. Other times, fear is a liar, confusing you until you can't believe your own eyes. But the very worst fear: that's the fear that destroys all hope." Leo, The Darkness Within.

Everyone has fears, things that frighten them, things that make it hard to do things. You could have a fear of bugs, or the dark, or rats and it's perfectly understandable and won't be that big of a deal. Most people have a reason for these types of fears and phobias, but this isn't about that kind of fear. This is about the deepest, darkest fears that reside deep inside you. A fear that you won't acknowledge, one that some people don't even know that they have. Everyone has a fear that hinders them, makes them rethink what they should do, makes them focus solely on making sure their fear never happens, even if they don't even know what that fear is, it's an unconscious fear that lives deep inside people.
The episode The Darkness Within shows the guys have fears, and this should help explain them.

Leo: Failure. This shouldn't be that much of a surprise, Leo strives to be the perfect leader, the perfect ninja, the perfect son. This can be pretty draining on someone, and he is prone to not succeed every time. After almost dying at the hands of Shredder with his brothers and Sensei and almost all of New York, in the episode, Exodus Part 2, he got pretty moody and we can see that he believes he has failed and it's his fault. It took a long time before he was able to find himself and get back to the Leo we all know and love. His fear of failure never left him, it's still there. The fear in The Darkness Within was shown in the form of him failing to save Master Splinter over and over again, only getting out of it when he continued to tell himself that what he saw was not real. Thus ending the illusion.

Raph: His Anger. Throughout the series, it's plain to see that Raph has anger issues, and he is trying to gain control of it. His anger affects him greatly, to the point where he almost severely injured Mikey with a metal pipe, in the episode Meet Casey Jones. He can get so blinded by rage that he doesn't know what he's doing, his brothers usually can keep him from doing anything too stupid. Sometimes he has to go out on his own to cool down. In The Darkness Within, his fear is portrayed as him being Shredder, he is his own worst enemy, which terrifies him. He works hard to control his anger and not become what he fears.

Donnie: The demise of his friends and loved ones. This one is pretty self-explanatory. It's something that was touched upon in the show, for example, SAINW, that episode sent him to a distant future where Shredder had taken over, and future him had been missing for a long time, 40 years, this episode made Donnie suffer, as he was forced to watch everyone he loved and cared about die, he watched his brothers of a different timeline get killed in front of him, he learned that Splinter was killed and as was Casey. Donnie felt helpless in these situations as he did in The Darkness Within, where he was forced to watch Angel, their human friend, fall off a cliff, he was unable to get to her in time to rescue her. Thankfully the Angel thing was just part of his fears happening in his mind.

Mikey: Betrayal. Sounds crazy right? Mikey is the fun one, the one that laughs at everything, screams like a little girl at scary stuff, the one who makes his brothers laugh and relax. Mikey fears being betrayed, even though he knows his brothers would always be there for him, this fear is constant. It doesn't show that much as Mikey tries to ignore it and not show it. In The Darkness Within, his fear was shown as Leo joining up with the monster, and Leo's left arm being replaced by the strange arms of the monster. Mikey's fear isn't touched upon as often as the others, but he does show to try to stick up for his family at any chance he can. This fear fuels a lot of what he does, because he does not want this fear to happen.