Technically a monogram requires more than one letter, a bit confusing since "mono" means one. A monogram requires two or more letters that make a new form, the letters must be so "entangled" that they cannot be separated from the whole. For instance the monogram on the right, SAB, is completely dependent upon all three characters to maintain its form. If one were to remove the "S", both the "A" and the "B" would lose their crossbars.
Today when we refer to monograms we are most often actually referencing a cipher which is two or more letters that are not dependent upon each other. In the example on the left, A & Co., If one were to remove any of the individual letters, the form would not fall apart. Sure the overall composition would not be balanced, but none of the letters would be "broken."
When a single letter is used it is simply called an initial.
How does one choose an initial for one's self or as a gift? There are a couple of rules of thumb to remember:
1) Young ladies usually prefer the initial of their first name. How does one define "young?" That's a great question! It's really not a question of age, but of marital status. If single, the person historically would choose her first initial; but even this rule gets a bit wonky with single women of a particular age.
2) Ladies who aren't "young," historically use the initial of their last name. Again, this has been associated with marital status, and not actual age; but this rule doesn't carry as much weight today as it did in the past.
The take-a-way here is do whatever you want! These are great guides to use if you're giving a gift, but don't get hung up on them, remember it's the thought that counts.
3) A woman who is getting married should not use her married initial until after the wedding. Even if she is sending a thank you from both her and her fiancé, she should use her first initial or the initial of her maiden name until after the marriage.