Nobody can really be sure who invented the first film projector. There are many claims and different fathers of cinema but I'm going to side with the Frenchman Louis Lumiere. Not only did Lumiere have one of the first widely recognized projectors but he also showed the incredible foresight that defines most Hollywood executives when he famously spoke, "The cinema is an invention without a future."
Lumiere's claimed date of invention was 1895 so let's say that the second century of cinema actually started 13 years ago. So which films since 1995 have proven to be the most influential? To start, there's a difference between influential and inspirational. Many people have been inspired by Little Miss Sunshine, Best in Show, or L.A. Confidential, but those films didn't really have a lasting impact on the fact on the face of the Hollywood. A lot of people cite "Lost Highway" as a classic but nobody has been able to (and few have even tried) to follow in Lynch's footsteps.
Similarly, a film has to have a lasting impact. Harry Potter and Lord of the Rings started an onrush of kiddie lit movies but almost all of them have been disappointing. My Big Fat Greek Wedding got studios thinking that there was money in independent film but now everyone's closing their indie arms (mainly because the faux-indies still held to the Major Studio way of thinking and would have had nothing to do with a film like Greek Wedding). Some might sing the praises of Sex and the City saving female-centric films, but we heard the same song and dance after The First Wives Club. In the end, you can't tell the true influence right away so for this list, I'm only looking at the first decade of this second century, 1995 - 2005.
1. Toy Story (1995)
The reign of Pixar and computer animation begins!
2. Independence Day/Twister (1996)
Bombast over believability, these films ushered in the new era of blockbusters in which special effects trumped substance and script. If it wasn't for these films, movies like Transformers might have to actually spend time on plots, characters, and a love story with more emotion than "I'm really glad I got in that car with you."
3. X-Men (2000)
Comic book movies weren't seen as legit box office threats in 2000. The marquee heroes (Batman and Superman) had fallen off, other attempts were either modest successes (Spawn, Blade, The Crow) or disasters (Barb Wire, Tank Girl). In Hollywood, X-Men wasn't seens as a property that had wide appeal or recognition. On top of that, the film was drowning in bad buzz. There had been multiple re-writes, it was over budget and the key role of Wolverine was re-cast almost a month into filming by some unknown foreigner named Hugh Jackman.
After X-Men became a surprise hit, any and every superhero film got greenlit and the Era of Comic-Book Movie began.
4. The Real Cancun (2003)
People think of it as a pipe dream now but back in '03, the reality movie was very close to becoming an actual genre. The Real Cancun was getting a major release and another spring break film from the producer of The Bachelor was in the can and about to get a legit advertising push. Reality was graduating to the big screen. Of course, the Real Cancun was a complete bomb which led to the other spring break film getting shelved, and the megaplexes were saved from the Reality Invasion. But just think, if more people had gone to see The Real Cancum, The Simple Life could have been a film and we’d probably be seeing a Hills movie this summer.
6a. Tommy Boy (1995)/Billy Madison (1995)
Old School (2003) helped launch a new batch of Coming of Age comedies but these were the originals that helped pave the way. Like Apatow's films, these movies were not about the troubles of being in a relationship as much they are about men having to grow up to even be able to function in the real world.
6b. There's Something About Mary (1997)
Hollywood learned that a) gross and stupid sell and b) Dumb and Dumber wasn't a fluke. The R-rated comedy regained it's footing in Hollywood with Mary as the Farrelly Brothers showed that gross-out comedies could out-gross even the highest expectations. They laid the ground work with Dumb and Dumber but Mary was a hit both in the box office and with critics. Two years later, the floodgates would open after American Pie brought the gross out gags to the teen comedy.
7a. The Ice Storm (1997)
The Problems of the Upper Middle Class becomes High Art. In the ultimate post-modern shift, the people who drown themselves in hipster chic to escape the problems in their lives suddenly make their problems into hipster chic. I think Ang Lee's outsider sensibilities made this film better than the new wave of more self-indulgent films (The Savages, Margot at the Wedding) but I believe that this film helped set off the trend.
Personally, I wish that more filmmakers addressed the personal demons/community lashing out at whoever they can sentiment of 1997's "The Sweet Hereafter" as it resonates more with the current political climate. Instead, American Beauty took The Ice Storm mindset to the Oscars and it's been boo-hoo bourgeois ever since.
7b. Happiness (1998)
Mainstream appeal becomes the enemy. Happiness made no attempt to be liked and, in fact, was almost revelling in the fact that it wasn't for most people. This film was instrumental in the break of indie films from the mainstream. Before, people would whine "Oh I wish people could discover the truly great films like Drugstore Cowboy or Welcome to the Dollhouse." After "Happiness", it was no longer a problem if the mainstream didn't like the movie; instead it was like a badge of honor. The film mixed John Waters's audacity with the standard Young American Angst, making it more palatable to the every-hipster.
8. Scream (1996)
The horror movie would be brought back to life, opening the door for countless teen slasher films, classic horror remakes, and the torture porn movement. Films like Saw, House of 1000 Corpses and 28 Days Later may have directed the horror movement in new directions but none of it would have happened if not for Scream.
9. The Matrix (1999)
After this film, suddenly every action star knew some sort of martial-arts. Stylistically, it turned a new (and now much copied) page in the history of action and sci-fi films.
10. The Sixth Sense (1999)
Obviously, this film made the twist ending and creepy little kids en vogue but possibly the more important influence this film had was that it helped bring spooky back into a genre that had been over-run by slashers. People remembered that there was more to ghost stories and scary movies than just the shock of someone sneaking up behind a busty co-ed with an axe. You could have a spooky film without being a "horror" film.
11. Lock, Stock, and Two Smoking Barrels (1999)
While Tarantino focused on paying homage to his favorite genres, Guy Ritchie helped invent a new one. As Ebert wrote, "'Lock, Stock and Two Smoking Barrels' is like Tarantino crossed with the Marx Brothers, if Groucho had been into chopping off fingers". While Tarantino's style (namely his wordy scripts and hip dialogue) had its moment in the sun, Ritchie's brand has become a staple of the megaplex. Just as the Marx Brothers influenced Looney Toons, Ritchie's film helped lead to the cartoonish actioners like The Transporter, Death Race, Layer Cake, Crank, Wanted, etc.
12. Friday (1995)/The Nutty Professor (1996)
After years of disappointing to middling box office returns from African-American-led urban dramas and rom-com’s, the new (and only) black film movement would be broad comedies.
13. Saving Private Ryan (1998)
Redefined the war film by making neo-realistic action the Gold Standard. More cynically, it masterfully mixed violence with a War is Hell message that made everything seem like it fit for a legit purpose and wasn't at all gratuitous.
14. Braveheart (1995)
The period epic becomes a viable genre again, opening the door for Gladiator, 300, and others.
15.Don't Be A Menace... (1996)
The precursor to the Scary Movie franchise, this film also was the unfortunate inspiration for Jason Friedberg and Aaron Seltzer who've littered the theaters with one horrific film after another. This film also started the upsetting trend of making spoof movies that are loosely tied together bits that spoofed many different films/people.
16. Analyze This (1999)
Slumming ain't so bad. If the great Robert DeNiro can do a comedy, why can't Joan Allen pop up in Death Race? While the indie film was becoming more and more resistant to even being acknowledged by the mainstream, many of its former darlings started willfully signing on for silly films. "Analyze This" showed that the crossing over could be successful both in terms of quality and box office. The quality of these crossovers has pretty much declined ever since but that's not stopping anyone from cashing those paychecks.
17. Everest (1998)
The film that put Imax on the map.
18. Remember the Titans (2000)
If you’re watching a sports movie, odds are that it’s going to be some uplifting story of hope and faith. You can pretty much thank this film for bringing back the "Brian's Song" sentimentality to the mainstream sports film.
19. Chicago (2002)/Save the Last Dance (2001)
These are the films that saved song and dance. Chicago is obvious because of its 171 million dollars domestic take and Oscar win but Save the Last Dance was the film that paved the way for the recent spate of dance-centric films. Moulin Rouge! (2001) also deserves credit for helping to bring back the musical.
20. Traffic (2000)
While it's hardly the first film with interwoven storylines, the approach was perfected in Traffic. Nowadays, it seems like you can't make a political film without mixing it up (Babel, Fast Food Nation, Syriana, Lions for Lambs, the upcoming Crossing Over).
Honorable Mention: The Usual Suspects, Bottle Rocket/Kicking and Screaming/Flirting with Disaster, Dark City/City of Lost Children, Babe.