The most basic way that Hearts of Iron 3 can be described to a newcomer to the series is a game where you change the course of World War 2. We are not talking in ways like Wolfenstein, where you are a lone agent who takes down a whole war machine. Rather, you are controlling the nations that participated in the Second World War on a micro-management level. The game, and the series in general, tries very hard to strike a balance between deterministic history (the computer sets you on a historical path and you deviate slightly) and dynamic history (you can dramatically change the course of the world).

In regards to history, Hearts of Iron 3 is not for those that seek to have a strict historical management simulation or a truly free and dynamic wargame. While you can change the course of your particular nation, other nations that you are not interacting with are relatively locked in their course. While you can shift things quite a lot through expert use of diplomacy and warfare, it is rather limited, which may disappoint those looking for a more open experience. There will no doubt be mods that make the game more deterministic or more dynamic, but at the moment, it strikes a balance between the two fairly well.

Hearts of Iron 3's gameplay is split into many different screens. These screens, which are more essential to your effort than anything else, are the main screen, diplomacy, production, technology, politics, intelligence and statistics. You will spend most of your time buried in the micro-management screens and less on the main screen except in case of major conflict, so do not be alarmed if you haven't seen your map in a while. It does not matter all that much. You can govern your country almost entirely without use of the main screen. If you are neutral, you very well could play it without the mains creen whatsoever.

The main screen acts as a gateway into the world of Hearts of Iron 3, and its interface has been greatly streamlined over its predecessor. It no longer takes up most of the screen, and it is more clear what specific buttons do. You may also notice something excellent in the main screen: there are many more provinces. In fact, each province from the second game has been broken up into four or more provinces to invade, and these original provincial distinctions are made more clear by using a specific map filter. While this makes the game more complicated, it also allows for much finer movement of troops than in Hearts of Iron 2, and that is a huge boon.

Hearts of Iron 3 lacks sprites, which is something else that differs between it and the second game. The only sprites available are US and German sprites, and only as a DLC pack. Your armies as they are represented in-game are boxes with formation and unit compositions summarized through a pixelated portrait. For example, Infantry might be an X, while militia might be a big M. It's sort of a cop-out way to get people to pay more for DLC, but when taking into account the massive lag issues the game already has, the abstraction of the armies is a welcome touch.

Diplomacy is much like it is in Hearts of Iron 2, but with a few minor changes. The first is that you no longer pay diplomatic points as much to instill actions upon other countries. Rather, you expend actual diplomats. The exception to this is the "Influence Nation" action, which pushes a nation towards your faction of choice (Allies, Axis, Comintern). The Axis and Allies factions are well known, with the Comintern being the communist and Eastern European forces lead by Russia and initially allied with the Allies. You will not be spending a lot of time in this screen, which is sort of sad because more robust diplomatic options would have been nice.
Production is not much different in Hearts of Iron 3 than earlier games. You determine what you want to build, where you want to build it, and how you want your industrial manufacturing power to be used. There is some differences in your sliders, and more noticeably, in how you build your divisions. You produce divisions by choosing the brigades that you want and then approving it for construction. For example, you can have a division consisting of one militia, one armored cavalry, and two standard infantry.
Hearts of Iron 3 improves greatly upon the model of research introduced in Hearts of Iron 2. In Hearts of Iron 2, you had advisers to hire who has specialties in certain areas which would translate into better research. Hearts of Iron 3 has thrown this model out the window for a far more intuitive "improve as you learn" method. Whenever you research something, it researches the technology and adds to your skill level in the field that the technology resides in. For example, researching Industrial Production increases your Theoretical Mechanical Engineering. These skills are divided into two camps: theoretical, which doesn't decay but can only be raised through research, and practical, which does decay but is obtained through performing actions on the battlefield, such as making your infantry units engage enemy forces. It sounds complicated, but you'll find it to be very simple and intuitive. Think of the Oblivion model of raising stats, and you have a good idea of how the system works. Easily the best improvement in the entire series.