Iaido is roughly translated as "the way of the quick draw". You can think of it as a class of different martial arts, not a style in and of itself. The actual style will have a proper name like Muso Shinden Ryu. So in that case, it is called Muso Shinden Ryu Iaido (but most people just refer to it as Muso Shinden Ryu, since iaido describes pretty much everything they do). There are many ryu that can be called iaido.
Each ryu that teaches iaido generally has a series of sword kata, both solo and partnered, that must be mastered by the student. There are beginning kata up to advanced kata. These kata teach what each ryu considers the correct and most important techniques and principles for actual sword combat. The movements themselves are often useless without a good teacher explaining what is happening in the motion and the philosophy behind it.
There's also something called "Iaijutsu". Whenever you see -jitsu or -jutsu, instead of -do applied to martial arts names, it implies that you are doing it for the purpose of combat. Whereas the -do arts imply that combat is no longer the central focus of the art. Instead, personal development (usually of ones character and spirit) is the goal, and combat is secondary.
Similarly, there is kenjutsu vs. kendo. Kendo is worlds apart from kenjutsu, by the way. The two often don't look a thing like each other. That's because kendo has been highly standardized and made into a sport, with rules and limits placed on what you can do, how you can attack, what you can attack, what you wear, etc. Whereas, kenjutsu schools teach everything relating to combat, are each highly stylized and different, and do not teach it for sport.
Your Kendo teacher is correct that Iaido practice will help your Kendo performance. The sword cuts and methods of drawing the sword that Iaido teaches will give you a more precise, more focused, and stronger technique as well as strategies for dealing with circumstances you'll find yourself in when doing kendo. And of course, there are many things Iaido teaches that do not apply to Kendo practice.
The feeling you get when practicing kendo and iaido will differ completely. Kendo will emphasize the reality of having an opponent who's there trying to win against you. You are going to be highly active, breathing hard, sweating, and fearing getting hit. It's a very good test of how you'll perform under stress.
Whereas iaido will feel like everything is frozen, calm, and without emotion. Everything is over in an instant in Iaido. The only stress you feel will come from your own mind.
So as you can see, you need both. Or actually, I'd say all three: Kendo, Iaido, and Kenjutsu. Kendo gives you a framework for seeing how your Iaido and Kenjutsu can work when you have to attack and defend simultaneously against an opponent who is trying his/her best to win against you. It is messy, interactive, stressful, and alive.
It's one thing to practice sword technique in a nice environment, in solo or possibly with a partner who just lets you do stuff to him. Yes, that requires great skill and a level of mental concentration that most people will never achieve. But it's quite another thing to have to deal with someone who's attacking you. Iaido and Kenjutsu give you the former, and Kendo gives you the latter. Each aspect gives you something.
No, Iaido is not necessary for learning Kendo. Plenty of Kendo students do fine at the sport without ever learning a single Iaido kata. (Actually, Kendo itself has kata which imparts some concepts that Iaido also imparts, but that's besides the point.)
And no, Kendo is not necessary for learning Iaido, either.
But each does complement each other. There's some overlap between them. The insights you get from one can help the other. But more importantly, combining all these aspects of sword fighting will make you better at sword fighting.
I think it's more rare that people train in Iaido these days, even and especially in Japan. Kendo, with its emphasis on the sport aspect, has really taken off in popularity. It's Japan's national pastime. But it's also not uncommon for Kendo students to train in Iaido or Kenjutsu to make their understanding of sword fighting more complete. It's well understood that Kendo is far removed from the sword technique trained in feudal ko-ryu.
Hope that helps.