Puppies whine and cry for various reasons. These include
Usually when people ask this question, it's because they're trying to get a good nights sleep. When you first get your puppy home, it helps to consider that your puppy is now in strange environment and probably been taken away from its litter mates and mother for the first time. New smells, new sounds, new sights, new people! It can all be a bit over-whelming for them.
Your puppy is lonely and afraid
Generally then, in the 1st few nights, if you leave your puppy alone, they are probably whining because they are afraid. They've never been on their own before. It's not a natural thing for a puppy to experience - dogs are social animals which means that they have evolved to be with other animals (or people). Before they get the opportunity to learn that it's ok to be alone, their little minds are reeling - "oh no, i'm alone, i'm going to die!!, help help"!
It can help to sleep in the same vicinity as your puppy for the first few nights. This not only helps with house training, but also helps them to settle more easily into their new environment.
Additional things that can help are
Do make sure that your puppy has had plenty of opportunity to get out and toilet before you leave them for bedtime. You may also benefit from getting up in the middle of the night for the first few nights to let them out - before they start whining. For more help on puppy house training, click here
Your puppy is trying to get your attention
Again, puppies like company, and they like fun. They don't like boredom. There are many reasons why they're trying to attract your attention - but generally what happens is that they very quickly learn that "whine = mum comes to see me"! Make sure that you don't allow your puppy to teach you to come when called (ha ha) - only go to them when they are quiet (unless ofcourse there is something hazardous or toilet-wise going on). If your puppy whines, then distract it with a calm sshhh, shhh noise or - clap your hands quietly..... this should quiet it for a few seconds. During the quiet time, go to your puppy.Your puppy is bored or frustrated
Make sure that you have given your puppy the appropriate amount of exercise, some good quality training, and good quality play - throughout the day, and before bed time. Don't get the puppy too excited - just a little calm play with some nice toys. Chew toys are great too to keep your puppy occupied for ages - do make sure that they are the right type of chew toy for their size and breed, and that they are safe.
Ofcourse, some puppies are just a real handful, and there may be many other things going on. If in doubt, get your puppy checked over by your vet for sound health, and ask them to recommend a qualified behaviourist / trainer to help you further.