First 100 days Party
In the old days, many people died before they were born, or before their first 100 days of life. Therefore, someone who lived past the 100-day mark could be expected to live a full life. This day is really more a "welcoming" day than a birthday and is now a tradition.
First Year Birthday
Similar to the 100-day celebration. So many people died before their first year of life, so if they reached one year, they were most likely to continue to live a normal life span.
60th and 70th Birthdays
Same as above. If someone lives to age 60, or especially 70, they celebrate their having such a long life.