Like Will and others have already said, the Western is too big to put into some simple cubbyhole. What are you looking for in it, or do you know?
Westerns don't have to be set in the old West--any frontier will do, especially space. Outland (1981) is High Noon set in space. Star Wars is a kissing cousin to every Gene Autry or Roy Rogers Saturday matinee Western where the good guys convinced the local townspeople to stand up to the bad outlaw gang. Avatar is basically Clint Eastwood's Man With No Name riding into town in a space version of A Fistful of Dollars, pretending to be on one side while helping another.
The ocean is another frontier that can stand in for Westerns. Is a film about people in an underwater city having to fight off sharks whose intelligence has been genetically enhanced all that different from a western town or outpost having to fight off Native Americans who have a superior knowledge of their environment and put that knowledge to use in their attacks?
Sometimes Westerns are disguised as latter-day dramas, as in Bad Day at Black Rock, which is a drama of prejudice and murder set in the modern west, but still a basic cowboy yarn at heart, with the stranger riding into town to right wrongs against all odds.
The biggest difference that I see in Westerns of today vs. the Westerns I grew up with in the 1940s-1950s can be demonstrated by viewing the original black-and-white 3:10 to Yuma (1952) starring Glenn Ford and Van Heflin vs. the recent widescreen, colored remake with Crowe. The original shows the stage holdup and killing, the capture of the head of the outlaw gang (Ford) and the stopover at Heflin's ranch where the outlaw is switched from the stagecoach to one destination to go by horse to another destination. At the Heflin ranch, the outlaw has supper with the family, including two sons, one preteen and the other in his early teens. The teenager in the original looks like a child, as did most 14-15, even 16-year-old boys in the 1950s. In the remake, the teenager is as tall as his dad and needs a shave. In the original, the boys are respectful of their dad (as most of us guys were back at that time), proud of his role in capturing the outlaw, but worried about the risk he runs in taking the outlaw in. Van Heflin's character mentions having fought in the Civil War but makes no big deal of it--as was the case of a lot of dads in the early 1950s who had fought in World War II. In the remake, the teenaged son is not only disrespectful but downright antagonistic to his dad who he knows (god knows how or why) shot himself to get out of frontline duty during the war. The dad is a coward and failure and therefore by modern standards there's no reason for any of his family to love and respect him or give a damn whether he lives or dies while trying to get enough money to sustain his pathetic ranch a little longer.
In the original film, no time is spent on the trail. They're seen leaving the ranch and arriving in town where they are to take the train to Yuma. Most of the film takes place in the small hotel room where charming bad guy Ford tries to talk Heflin into taking a bribe to let him walk away instead of dying in the dirt in an unsuccessful attempt to take him to the train. Gunplay is at a minimum; dialogue, drama, and suspense are at a maximum. The remake lingers on the trail where they encounter such PC images as dishonest whites taking advantage of the poor ignorant Chinese, and a shoot-em-up with warring Indians in which Crowe demonstrates his superior abilities. It's all action and MTV fast editing.
In the original, stageline-owner Butterfield manages to hire 3-4 men to help stand off the outlaw gang, but when the outlaws threaten "quit or die," they quit. In the remake, the outlaws say they will let them walk away but then gun them down for no reason, although it's "common knowledge" today that killing one terrorist will just make 3-4 more from the family members he leaves behind. If the people are willing to quit, what besides a street-gang mentality would justify killing them?
In the original, people won't help the good guys, but they don't help the outlaws either. In the remake, the outlaws (having just gunned down 4 guys they promised to let go) now tells the town they'll pay them to shoot the good guys, and of course the townspeople start blasting away, burning more powder than during the entire Civil War. Illustrating, I guess, the modern conviction that the average people on the street will do anything--even murder each other--for money.
In the original film, Heflin does the right thing, succeeds with the assistance of Ford in another version of The Code of the West, the rains come, everyone is happy with Ford convinced he'll again break out of Yuma prison. In the remake, the father makes a half-ass but ultimately unsuccessful attempt to get Crowe on the train and gets killed in the process. But then Crowe for no apparent reason, kills his own gang and takes himself to prison. What's the point of all that?
Basically, the main difference is that the original film is shaped toward the entertainment values and mores of the 1950s whereas the remake is shaped for the entertainment values and mores of the 21st century, including the equivalent of a "drive-by" shooting where one outlaw dashes on horseback down mainstreet and pots one of the captors of his boss. The gunfire and body count is ramped up in the remake to meet the expectations of the 15-35 year-old-males, or whoever they make movies for these days.
That age group much prefers the remake whereas guys my age much prefer the original.