So! First off, no more peeling! Your nails are not tools. That goes for peeling your nail polish off as well, peeling removes layers of your nails which makes them more thin and brittle. (Also easier to break!)
You should also file and push your cuticles back at least once a week. Pushing your cuticles back helps promote nail growth, also making life a lot easier when you do your manicure, seeing as your grown out cuticles won't be in your way. Filing them regularly, as opposed to just clipping them with a nail clipper, makes them stronger. (And harder to break!)
When you take your polish off, try this one cotton ball method. Unroll your cotton ball and take little pieces (the size of your nail) and dip them in a little polish remover, set over your nail and continue with the rest of your nails, let the remover do the work, it's designed to do so, right?! Once you get to the last finger, you should be able to go back to the first finger and gently apply a little pressure on the cotton and pull toward you, removing the polish with it! This removal method is great for so many reasons, little work for you and you'll save tons on cotton balls and remover! This method is also better for your nails then scrubbing at them with the cotton ball, that actually does damage on your nails so, the less scrubbing, the better. Sometimes you'll have a little polish around the edges still, I just take another piece of cotton and clean up where it's needed.
ALWAYS use a base and top coat when you do a manicure, it helps protect your nails when you remove your polish by making it easier to get off and also keeps them from getting stained, nobody wants that! A top coat will seal everything in and make your polish last. Also be sure to use cuticle cream after your mani is complete! Healthy cuticles means healthy nails!
I used to have really thin, brittle nails and they broke and peeled constantly. Just doing these few simple steps has made a HUGE difference! My nails grow a lot healthier and faster, it's awesome!