Most adventuring parties I've been in or run have had four or more people. There's even a few games I've been around where there was more than one GM. That's not always an option, though. Sometimes you've got very few other gamers in your area or schedules just don't line up too great. Maybe you just prefer a smaller group. A far extreme is for there to be one player and one GM. This has it's benefits and it's drawbacks: the game can be more easily tailored for the group's tastes, but there are fewer influences on the story. Rules are understood only as far as the two involved grasp them. A trade-off occurs often between party size and how alive the characters are, but there are fewer engagements to arrange around for the whole group to game.
Any GM knows that when they find what their players want, giving them more of it will make them happy. It's a fun trick-of-the-trade. We also have things we GMs enjoy, so you can expect to see some or a lot of those in there. Mostly the game tends to go towards the party, though, because there's usually more players than GMs. When there's one player and one GM, though, the world takes on the life which both players want it to have. This is emphasized more strongly with smaller groups because the tastes of others dilute the enjoyment derived from the game less. I'm not saying I don't enjoy what my friends bring to the game table, I'm saying that not all of the things they find fun are shared by me. Fewer preferences around the table means more customizability.
At the other side of the coin, fewer voices are around the table means a smaller number of forces influencing the story. Even with a great player and a great GM, this can bring a game to a standstill. The variety of diversity stirs the cauldron well where role-playing games are concerned, especially story and plot. Character development, too -- if there's a greater number of people with varied characters then there's more of a chance to learn how your character responds to various external stimuli. Otherwise it's all on you and the GM to influence the story, exclusively.
This next part can be either a boon or a bane: rules for any given system need be understood only among the two people playing. On the one hand, that suggests fewer rules needing to be researched if fewer characters are at the table. If your player is running multiple characters, however, then odds are they might be operating one or many who fall outside their typical specialty. This isn't just for novice gamers, either -- all of my time spent playing an arcane mage did little to help me understand the variety of divine spells in a particular system. It's really touch-and-go whether rules knowledge will be helped or hindered by having so few people at the table, but it definitely is impacted by whether or not the player has one lone character or a party at their command.
Regarding that difference, a large trade-off often occurs when players take on multiple characters. Character development works best when we can focus on that one person, feel them out and explore their reactions to different situations. It's not the only way to do it, though it's certainly the most focused way. Role-playing two characters simultaneously often takes on a very mechanical feel as a result. It's not writing them out or planning them which suffers, it's the real-time role-play that takes the hit. While that isn't the case all of the time it's certainly the case most of the time. An alternative to this is simply to play fewer characters, and perhaps even focus the story on a single character. Hopefully that player doesn't mind making their character do whatever it takes to survive in the adventure which they and the GM are having; specialization might have to be put on hold so they can learn to fight, for instance.
One of the greatest benefits to gaming with one PC and one GM is that games rarely ever have to go on hold. Maneuvering schedules for four, five, six different people can be quite the daunting task. Some GMs I know of delegate that task to a player because it's such a chore for them. It stands to reason that in most cases, fewer schedules to juggle means more opportunities to game. This is the biggest reason I opt for solo games, when time permits. To date, the game I progressed the farthest in was a home-brew in which the GM and I spent at least eighty percent of my character's career alone, with no other players at the table. Even as that friend and I have grown apart, the memories we made together over that game haven't ceased causing me to smile five years after the fact. It was a blast, and I hope that each one of you gets a chance to experience something like it. Happy solo-ing!