Watson makes an inference about the client, and Holmes makes an inference about the letter writer’s handwriting.
An inference is a guess based on evidence. When you make a guess you almost know for certain, because it is an assumption backed up with facts.
The client, Mary Morstan, is looking for her father. In this chapter, both Holmes and Watson make inferences.
Watson, you will note, makes some inferences about Mary Morstan herself. He is quite smitten with her! (Pay attention, you will learn what happens between them later.) She dresses in perfect taste, but he infers from the way she dresses that she has “limited means” because she is wearing simple clothing. In other words, she is poor. He also infers that she is upset.
I could not but observe that as she took the seat which Sherlock Holmes placed for her, her lip trembled, her hand quivered, and she showed every sign of intense inward agitation. (Ch. 2)
This turns out to be accurate. She is upset. Her father appears to be missing, and she is hiring Sherlock Holmes. Most of his clients probably are agitated.
She brings Sherlock Holmes a series of letters written in a disguised hand she says is not her father’s. He makes an inference from the handwriting. Watson believes that the man is a man of business and character, but Homes disagrees.
Men of character always differentiate their long letters, however illegibly they may write. There is vacillation in his k’s and self-esteem in his capitals. (Ch. 2)
Holmes then decides to go out and do some more investigation. As usual, he doesn’t tell Watson any more than that, and he and the reader are left hanging.
You will notice that Watson’s inference was wrong, and Sherlock’s was right. At least, he says that it was right. We will have to wait for other events to unfold to see if it was. (It was. It usually is.) This is the case with their relationship. Watson makes a deduction, sometimes even at Holmes’s behest, and then Holmes explains why he is wrong. Holmes is the genius, and Watson is the plodder. Watson, the narrator, is always a few steps behind Holmes, so the reader never really knows what is going on. That is the genius of Doyle. The reader can try to guess, but never really has all the clues.