You generally would breed hamsters according to their specific breed ie Breeding only Syrian hamsters or breeding only Chinese hamsters. It is not advisable to breed hamsters of different breed types. If you've got a friend who has a hamster of the same breed of the opposite sex, you can arrange with him/her to breed your hamsters and then share the load of having to find homes for the young. This is the case for breeding Syrian hamsters as they are more often than not kept in isolation. Dwarf hamsters are different in that they are sociable and can live in social groups - so long as they are introduced to each other after weaning and they 'grow up' together. If you have a cage of dwarf hamsters (who are not of the same sex), they will start to breed when they are ready - and they generally will pretty much 'self regulate' their breeding pattern and frequency and you just have to 'watch' it unfold.
Generally for female hamsters, the ideal age to start breeding is at 4-6 months - any younger and you would risk her having a poor litter (she has still to grow herself) and also increase the likelihood of cannibalism. Leave it any longer than 6 months and you run the risk of birth complications. For male hamsters, they are sexually active from as young as 6 weeks of age but it's probably best to leave them alone till they're about 2-3 months old before contemplating breeding from them. Dwarf hamsters kept together will start breeding at 2-3 months of age.
Whilst you can just keep her breeding litter after litter, this will greatly shorten her lifespan. It's best to wait about 3-4 months before you breed her again. This will give her body time to 'recuperate' and put her in a better condition come mating time. If you stick to this regime, it will mean that you will probably get 2 litters out of any one breeding female hamster as they often become sterile when they turn 14 months old. Sometimes you can get a third litter from a female but this is the exception rather than the rule. Strangely enough, this is the self same pattern that dwarf hamsters will adhere to even though they have 'free rein' as to how often they mate and produce offspring. 2-3 litters per breeding female is the norm for dwarf hamsters kept as mating pairs.
Because Syrian hamsters are kept in isolation, you usually have to play 'match-maker' when it comes to breeding frivolities. Female hamsters come into 'season' once every 4 days (ie they ovulate during that time) and it's only when they are in season that they are 'receptive' to the advances of the male hamster. Any other times fighting would inevitably occur.
Given that they will go into season once every 4 days, if you get it wrong the first time, there's always the next time. Generally, the female hamster will emit a musky odor just before she comes into season and inevitably, it's always around evening time that the 'oestrus period' (time when she's fertile and accepting of the male) starts and it lasts for anywhere between 4 and 18 hours. It is during this 'window of opportunity' that she will be receptive to the male so breeding them has to take place then. After the female hamsters finish their season, there is a thick white mucous discharge that is excreted from their vaginas. Less experienced hamster breeders will then take that as a starting point, count another 3 days after that and then try for a breeding encounter (anticipating the next oestrous cycle 4 days later). If the weather is cool (ie winter), most hamsters will not breed. The other tell-tale sign that she is in season is that when you stroke her back towards her tail, she will freeze on the spot, flatten out and lift her tail.
Okay, you've established that the female hamster is ready for breeding, what now? You have 2 choices - you can put her in the male's cage or take them both out and put them in a separate breeding box (neutral territory). Wear protective gloves, have an object ready to separate them if any fighting ensues - they often aren't very selective about who they nip and will nip you in your efforts to separate them if they choose not to breed. If you've picked the right time, the female will 'freeze' with her tail in the air and allow the male hamster to mate with her a few times. Keep them together for anywhere between 15 minutes to an hour - when they start to lose interest in the mating ritual, then it's time to separate them. Virgin males can be a bit confused initially and may turn the female hamster off by mounting her from all directions but they will soon learn 'on the job'. Breeding is generally more successful when both male and female hamsters have had previous mating encounters.
If the female doesn't come into season again 4 days later, she's regarded as being pregnant. Given that the babies only stay in her for 16 days, 10 days after the mating, she will start to become decidedly pregnant - her abdomen will swell (no, she's not getting fat), and her nipples will start to become more pronounced.
You return both hamsters to their respective cages. Female hamsters will need extra food during the pregnancy. High protein foods like tofu, boiled egg, wheat germ (to aid in birthing) can be added to the diet. She can still be handled but it's best to leave her alone when it's closer to the expected birth date. Clean out her cage at day 14 and take out the exercise wheel, provide her with extra nesting material (eg shredded paper). Then put her cage in a warm, quiet spot with as little disturbance as possible and wait...
Let her nurse the babies till they are about 3 weeks old. DO NOT disturb the nest during this time as she may decide to turn on them and eat them. The cage will smell during this time - that's unavoidable, any attempts to change the bedding material would increase the likelihood of cannibalism so it's best to just leave it for the moment and put up with the smell. Sometimes despite all this, the female hamster will still cannibalize her young, this is common in female hamsters bred under 4 months and in older females who will cull some babies to reduce the numbers in the litter. Continue with the high protein foods (other ideas are bread soaked in milk) during this time. At about 3 1/2 weeks of age, the hamsters are ready to be sexed and weaned - they are no longer reliant on their mum for nourishment. You can keep them in 2 different boxes/cages (all the male hamsters in 1 cage and the females in another). They can be kept in these groups for a further 2-3 weeks. In the meantime, you can find new homes for them and they can go to their new owners at 5-6 weeks of age.