When a person does this it means he/she is avoiding you, right?

If it is a stranger, it may mean they are avoiding having to make eye contact and acknowledge you.

If it is someone you know, it means they’re avoiding you.

Of course it’s possible this person was avoiding you. It’s equally possible that they didn’t see you coming towards them in the first place and so wasn’t paying attention.
If there’s a bone of contention between you two, it might be the former

If I’m thinking hard about something while I’m walking I could pass my daughter on the street and not notice her. If she were to speak to me I’d scream and leap with surprise, be glad to see her, and feel really silly. Sometimes people just don’t register what’s in front of them, if it’s not in normal context.

One thing it could say is that the person is really nearsighted.

Someone whom I was just getting to know and like made it a point to explain to me that he would probably not recognize me (or anyone else) from more than a few feet away and that I should not take it personally but rather come right up to him and greet him at close hand.

If it’s someone you know, they could be upset about something, and in no mood to chat.

Avoiding you? Maybe. There could be a hundred other reasons for it. Maybe they’ve forgotten your name and don’t want to embarrass themselves. That’s usually why I do it. I forget the names of people I’ve been working with for years. I have numerous clients who I knew pretty well, and I just can’t bring their names to mind. I’ll often pretend I don’t see them.

I do that to everyone because I have a strange feeling that it would creep them out if I made too much eye contact.
Basically I’m nervous.

If you know that person they are either avoiding you or didn’t recognize you.