Puppies go through a critical period of socialization between 8 and 16 weeks of age. During this period of time they mature very rapidly. If isolated from external stimuli and not exposed to a wide variety of sights and sounds, they can grow up to be fearful adults. HOWEVER, during this time they are losing immunity from their mother and are vulnerable to disease before their vaccination schedule is finished.
Here's how this works:

A YOUNG PUPPY IS IMMUNE TO NEARLY EVERYTHING

If the mother dog was vaccinated and/or exposed to diseases, her disease-fighting antibodies (in the form of protein molecules) are passed along in the first day's milk, called colostrum. Some breeders re-vaccinate just before breeding to make sure the mother's circulating antibodies are running high. These maternal antibodies last for many weeks, while the pups are at their most vulnerable.

Only the first 48-hour's milk (colostrum) contains maternal antibodies, so how long the puppies nurse after that is irrelevant. Also, the more effective digestive system of older pups breaks down the large protein molecules of antibodies, so it is useless to try to give older pups colostrum "boosters".

Creating a NEW immune system

Eventually, the maternal immunity wears off. The pup has to form its OWN antibodies by direct exposure to pathogens. Since several very common diseases have very high mortality rates, we would prefer this to happen through vaccines. Vaccines inject 'dead' or partial pathogen for the immune system to learn to recognize and fight. HOWEVER if there are still high levels of maternal antibodies in the pup's system during vaccination, they will 'kill off' the vaccine before the pup's immune system has a chance to respond. Normally, MATERNAL IMMUNITY WEARS OFF anytime from 6-18 weeks. Since we have no idea when that will be, shots are given every 2-4 weeks. The goal is to deliver a set of vaccines just as maternal immunity wearing off but before the pup is exposed to the actual 'live' disease.
One study of a cross section of different puppies showed that at six weeks of age, only a quarter of the puppies tested were able to respond to a particular vaccination and form their own antibodies. The number of puppies responding to the vaccine increased with age:
9 weeks, 40%
16 weeks, 60%
18 weeks, 95%
Other studies, with other types of vaccines, showed nearly 100% of puppies developing antibodies of their own as young as 12 weeks. So it really depends on the individual, their environment, and the vaccine.

HOW CAN I KNOW MY PUPPY IS SAFE?

To actually KNOW when your puppy has formed it's own antibodies against the most deadly diseases would require a panel of titre tests. This is quite expensive. Instead, most owners stick with the puppy-shot schedule and practice good sanitation during the "uncertain" period between 6-18 weeks.
Many canine diseases are common in the wild and easily spread by wild animals, cats, and even human foot-traffic. The pathogens can survive on surfaces for days and in soil for years. Also, other dogs and puppies can be contagious without showing symptoms themselves.
To protect a young puppy:

DON'T

  • DON'T Take them to grass or dirt areas with lots of animal traffic, such as trails, beaches, dog parks
  • DON'T Let them lick/eat off highly contaminated surfaces like the floors or counters at vet's offices, pet stores, or paved trails
  • DON'T Let them play with puppies or dogs unless you know the other dog is also kept away from unsafe conditions
  • DON'T Let the puppy swim in any body of water you wouldn't drink out of

DO

  • Clean visitors' shoe-soles with bleach
  • Wash hands and change clothing if you have been in contact with other dogs, even if they didn't look sick
  • Clean all used dog equipment such as toys, crates, kennels, and bowls with bleach
  • Wash hands, wipe paws when returning from an outing
  • Socialize at a class that disinfects floors and enforces clean shoes/paws

A video about socialization