If you want your eyelashes to curl, you have to train them to, which means using your eyelash curler every day. The more you use it, the less pumps you'll need for your lashes to stay curled. 

With that said, there are tons of ways to curl your lashes wrong. Here are some tips I've gathered from top makeup artists: 

• Always use a metal, high-end eyelash curler. With eyelash curlers, quality really does matter. Plastic curlers aren't strong enough to curl your lashes, so you'll be holding down on your lashes for longer periods of time. I'd recommend Shu Uemura, the queen of lashes, or Shiseido.
• Be diligent with replacing your eyelash curler pads. in general, you should be replacing those pads every 2 to 3 months. But if you start to see a dent in the pad, it's time to toss. Otherwise those dents will cut your lashes, which is exactly what you don't want. 
• If you have weak lashes, don't curl them after applying mascara. While pros often do this, they're called pros for a reason. Mascara can make the lashes stick to the curler, and when you pull the curler away, you may effectively rip your lashes out. 

Hope this helps! 

The previous responder wrote an excellent post. I would like to add one bit of info - if you repeatedly use curlers incorrectly then yes, then can damage the eyelashes - such as putting a ridge in them. However, this is not permanent and will go away when your eyelashes grow back. If you choose to gently heat your eyelash curler - as many recommend - be sure to test the temperature on your hand and keep away from the eyelash roots.