Hi, Mary Ann,
When a pattern has directions like those you asked about: K1 (k1, p1, k1) into next stitch, it is to give you the directions for variations on the pattern. So, whatever you are making probably has different sizes or some kind of variation.
An example would be a hat pattern I am working on. It says it's for Men (Women, Teens, Toddlers). So each step has instructions for all four. Like it says cast on 132 sts (120, 108, 102). So, I cast on 132 sts for a man's sized hat, 120 sts for a woman's, etc.
So when it sais K1 (k1, p1, k1) you are only doing one of those into the next stitch.
Does this make sense?