Well, for starters, the comment about a character being "one dimensional" is often just a derogatory term and a quick, easy, and lazy way of saying, "I don't like the character." In other words, the person hasn't even bothered to look into what one, two, and three dimensional characters actually are. They are just saying it to sound smart when they're clearly ignorant.
Anyways, the definitions of these terms does change a little depending on who you read. One writer may reduce all character to one and three dimensional characters, while others will claim that we should have three dimensions to characters. I forget where I read it, but I think the best explanation is this:
One Dimensional: Comes in, comes out. There's little, if any, meaningful interaction with the world, and obviously no character development. This would be equivalent to a character who gets a few lines in during a conversation but never returns, or they make only brief appearances later.
Two Dimensional: Has some interaction with the world, but not to a significant degree. They may have some character development, but they fail to have any serious internal or external struggles that seriously changes them and the world around them. Most people are referring to this category when mentioning one dimensional characters, provided they even understand what they are talking about. This may be a character who gets a little more attention, and they likely have a distinct personality. However, we know very little about their personal ideals, and if we do, they are often never confronted with dilemmas regarding their ideals or given a chance to seriously change as the story progresses.
Three Dimensional: Has meaningful interaction and development throughout the story. We know a lot more about this characters' personality and morals, and sometimes even go into why they are like that. They are often confronted with physical, mental, and emotional challenges relating to their limitations in those areas, and they are forced to confront potential flaws in their personality, beliefs, etc. This confrontation comes in a much more meaningful way than the other two dimensions, and we are often wondering if they will conquer or be conquered by the struggles that exist beyond physical ones. Often, these struggles aren't mutually exclusive and a physical struggle may be an attempt to show an internal struggle. Obviously, these are the most well-developed and often most remembered characters, not to mention the ones that offer the most discussion value.
Now, here's the thing: None of these character types are actually bad. Some characters are only important enough to the story to be one dimensional, and a lack of them may make the world feel slightly artificial. The same goes with two dimensional characters. Also, a three dimensional character isn't necessarily a good character. Their struggles may seem artificial, formulaic, paced poorly, and just there because they had to develop some way over the course of their time in the work. Ultimately, the only way a dimensions should be used as a criticism is if that character receives more time than their dimension indicates. For instance, a character that is present throughout most of the work but never develops beyond a two dimensional character, making them flat and forgettable. Otherwise, these are just descriptors, used to determine exactly what type of character the person is.
As far as four dimensional goes, I've never really looked into it, and this might even be the first time I've ever heard of the term. I don't see how you can go beyond a three dimensional character. I guess that if you want to further divide the characters you could get a fourth dimension, but I think that is rather pointless.