Fallout New Vegas ... is not sequel to fallout 3, and it doesn't continue along the story of fallout 3 instead it takes on a new one, however it does start in a time ahead of fallout 3. Fallout New Vegas is not considered to be fallout 4 of the fallout series.
[no spoilers here]
Fallout New Vegas starts out in the Mohave wasteland, a dry desert with very little fresh water and lots of creatures that are hostile. Fallout New Vegas's texture quality isn't as good as its predecessor, Fallout 3, and its map is a little bit smaller.
Nevertheless its still fun to explore the wasteland and meet new people. Fallout New Vegas has without a doubt much more "Stuff" in it than fallout 3 such as crafting and an expanded variety of weapons that fallout 3 didn't have, as well as double the number of side quests that you can complete. Fallout New Vegas has a very unique feel to it, it's map locations are usually very different, from underground jungles and mutated plants that can kill you all the way to a mystery city untouched by the nuclear fires of war with casinos and mafias.
There is still the karma system in the game which dictates whether you are a "good person" or not through your actions and deeds. But this is Fallout New Vegas isn't it? Yes it is, and now there is also a Faction system which tells you how liked you are in a certain group or organization. You can also be hated in populated areas, so yes people will kill you if you act like a jerk around them and commit crimes against their morals. Of course this is completely dependent on the group of people you join, as well as the type of person you'd like people in the game to see you as.
There's also a new feature called "Hardcore Mode" which I extremely recommend, since its not actually "hardcore". This mode will need you to satisfy your daily needs such as food, water, sleep, and your "fix", but that will only be if you get addicted to chems and alcohol, which are dangerous ... but helpful. Be warned though if you get addicted and suffer from a withdrawal, you will definitely find some side effects you don't want.
[might be some spoilers] [you have been warned]
The story starts ... with you getting shot in the head. THE END ... just kidding, but seriously you do get shot in the head. You act as a courier, a person who acts as a mail man, and without realizing it carry a very valuable item. A group of bandits and their leader capture you and take the said item after that you plan to take back the item and kill your killer, but I guess you learn that the leader is some hot shot from a big time place, and then stuff gets complicated.
You end up sending zombies to the moon, electing a new sheriff, killing legionaries, and such. But enough of that you play the game yourself. After getting shot in the head, you wake up in a place called GoodSprings and from there you start your journey.
PS
You can find a dead Indiana Jones in a refrigerator close to where you start as an Easter Egg.