寝る「ねる」means basically, 'to lie down' where as 眠る「ねむる」means 'to sleep/act of sleeping'. For example, think of it as ねる means to lie down, say in bed, whether asleep or not. But ねむる means to be asleep, performing the act of sleeping. There is also another word, 寝付く「ねつく」 which means 'to fall asleep' and refers to the act of actually falling asleep. I hope this has helped a little. XD