Measles is a highly contagious disease that can have serious outcomes including disability and death. However, in the modern era of medical care, very bad outcomes are fortunately not common. Most will recover and do OK.
The standard for immunization against measles is 2 doses with the first dose at 12-15 months of age and the second dose at 4-6 years of age.
The reason we have 2 doses is that a small percentage of people do not get a good immune response to the first dose. After one dose of vaccine, approximately 93% of recipients will be protected. After 2 doses about 97% are protected. So most are protected after a single dose but we get better results with a 2 dose series.
There is not a recommendation at this time to give an early dose of MMR vaccine to infants less than one year of age. Maternal antibody confers some protection up to about 6 months. That antibody also interferes with immune response to vaccinations for MMR.
For babies at high risk, either directly exposed or traveling internationally, there is an option to get vaccinated between 6-12 months for some protection. The issues with this are that the child will still have to get an additional 2 doses of MMR vaccine because this early vaccination does not count. Also, there is evidence that the overall immune response after all MMR doses are administered is not as robust when the early vaccination is done.
For children who have had their first dose of MMR, there is not a recommendation at this time to give the second dose early. Most have protection. However, if it has been at least 4 weeks since the first dose of MMR vaccine, they may receive the second dose of MMR to complete the series.
Adults who were born before 1957 are considered immune because disease was widespread at that time. Other adults should have documentation of 2 doses of MMR vaccine to be considered fully protected.