The Meiji Emperor would not have been called by that name until after his death. Meiji was his "nengo", or reign-name, and that name for Japanese emperors is only used after their death; e.g. the Emperor Hirohito is now known as the Showa Emperor, Meiji's name was Mutsuhito, and he was referred to as Emperor Mutsuhito during his lifetime.
The armor the samurai wear is from the mid- to late-Sengoku Jidai era in Japan, over 250 years before the Meiji Jidai era the film takes place in. This was deliberate (but still an error). Costume creators were aware that at the time the movie takes place, no armor would have been worn. They chose to have them in armor to show that the band of samurai wanted to go back to old values and rejected modernity.
When Algren and Katsumoto are shown in slow motion, during the horseback charge, Algren draws his sword and we hear the sound of the sword ringing as it is drawn. The Saya (scabbard) of the Katana is typically made of wood, and though they sometimes had fittings made of metal around the Koiguchi (mouth of the scabbard) these were usually ornate, and there clearly are none on the sword that Algren has.
At the beginning of the film, the narrator says a "sword" was dipped into the sea and the four islands of Japan were formed. This is factually incorrect; it was actually a jewelled spear, called Ame no nuhoko and is detailed in the Kojiki, the book of the Shinto faith.
After Katsumoto and Algren meet with Colonel Bagley and Omura before the final battle sequence, Algren rides back into the Samurai front lines, when he dismounts his horse you can see the horse kickback and hit one of the Samurai who then stumbles backwards unsure of what just happened.
(at around 1h 50 mins) After the generals meet in the field on horse back before the final battle begins and return to their own armies, Tom Cruise arrives back into the group of extras and as he's getting off his red haired horse, the horse kicks one of the extras (with it's back left leg) in his groin and the extra seemed surprised and confused on how to react.
Colonel Bagley mentions that the new Gatling guns possessed by the Japanese Imperial Army are capable of firing 200 rounds per minute. Unless the "new" guns are very early versions of the Gatling Gun (this is not what is shown), his statement is incorrect. By the 1870s Gatling Guns had a rate of fire of at least 800 rounds per minute and, depending on the specific model, significantly higher rates.
When Katsumoto is in the courtyard discussing poetry and perfect blossoms, he has a cut on his forehead from the ninja attack. He announces that they will be departing the next morning, but when we see him the next morning, the cut has completely healed.
During the final charge of the Samurai, Algren has his sword in a "stabbing" grip, ready to throw it. A few shots later, he has the sword in its regular position, flipping it to the throwing position again.
When the ninja invade Katsumoto's village, a ninja is shown quietly killing a samurai sentry by snapping his neck. Later, just before the final battle, the same samurai is ready for battle and staring the enemy army down.
When Algren is rescuing Katsumoto from his "home in Tokyo", one of the soldiers from the first group of soldiers that get a chance to fire has an arrow protruding from his back, with no ill effect, before the first arrows are loosed. It is clearly visible through the smoke from the guns.
During the wooden sword fight in the rain, (after Nathan spars with one of the young boys), in the first sequence, the sword is struck from his hand and flies away. As he stands up from the blow without moving to retrieve it, he has it in his hand again.
During the ninja fight when Katsumoto is at a stand off with two ninjas, one ninja has a sword and the other is holding two sai. In a close-up of the ninjas, both are holding swords, and then in the next shot of both ninjas, they have their original weapons again.
In the beginning, when Algren is doing the performance for the Winchester company, he fires a bullet at a bell. Before shooting the bell, he blows three whistles off of a calliope in the back of the room. When the camera does a close-up after he shoots the bell, only two of the whistles are missing.
In the scene where Katsumoto is escaping over the bridge and his son is providing cover fire with his bow, when Nathan Algren begins to run over the bridge with Katsumoto behind him you can clearly see Katsumoto's son still firing without any arrows.
When Nathan walks into Taka's house after his first round about town since his capture, Taka cleans up after him because he has walked in with muddy shoes. The shot before this showing him walking in the door clearly shows clean shoes.
In the middle of the movie, when Algren is experimenting in his new robe, the boy is seen watching him and begins to bow after being noticed by Algren. In the next shot, the boy's head is suddenly upright again.
Nathan Algren's pistol switches at various times from being a Colt 1860 with a Mason-Richards cartridge conversion to a Smith & Wesson Schofield. Since he is a recent veteran of the 7th Cavalry, the Schofield is the more likely weapon for him to possess.
The horse ridden by Captain Algren as he enters Tokyo with the Samurai band is not the same horse he was mounted on in the previous scene as they leave the village (the distinguishing mark on its forehead gives it away). Katsumoto's mount in this scene is not the one we usually see him astride in the rest of the film, either.
Japan *did* seek military advisors in the latter half of the 1800s to form a modern Army. The only problem with this is that they didn't consult the Americans to assist them. The most successful army at that point was the Prussian (not yet German) Army, whom they recruited for training purposes... as well as British naval attachés to assist in the creation of a modern fleet (which thoroughly embarrassed the Russians at the Battle of Tsushima in 1905 and established Japan as a fledgling world naval power).
In the final battle scene, the Gatling guns never traverse (move back and forth from left to right), but the bullets they fire, appear to spray all over, killing warriors who are scattered out across the field. Firing straight ahead as the gunners appear to be doing would not have hit the warriors out of the field of fire on either side.
At the final battle Algren tells Katsomoto about the battle at Thermopylae, and says that 300 brave Greeks held off an army of 1 million Persians. This is only partially true; there were 300 Spartans, but the Persian invasion force is believed to have had 150,000 soldiers. Of course, stories from ancient times describe the brave Spartans as heroes who stood up against 1 million Persians, and at the time of the movie it might still have been thought so. In addition, while the 300 were killed by the Persians (as they knew would happen), their stand inspired the Greek states to rally against, go to war versus, and eventually prevent an attempted Persian conquest of Europe. It's not unreasonable for Algren to describe the Spartan stand as a victory, not least because of the parallels between what happened at Thermopylae and the samurai's willingness to die in service of the Emperor - along with the fact that Katsumoto's stand does result in posthumous victories (the American treaty is rejected and the Emperor angrily confiscates Omura's entire fortune).
During the ninja attack in the house when Nathan and Katsumoto are fighting inside the house together. One of the first ninjas on the right hand of the screen that was injured or killed during the attack is seen rolling out of the way for the second group of ninjas to jump through the wall and attack.
When Algren Tom Cruise fights Higen in the rain, the bokken or wooden swords used are much too long for either child. The Nihonto (Japanese Sword) was made to correspond to its users height and arm length. This being measured by holding the sword in your dominant hand and letting it hang at your side. The tip of the sword should be just an inch or so short of the ground. However, it seems this "mistake" was deliberate as Algren would NOT have been able to fight with a sword that was the length either boy would be using.
When Katsumoto, Algren and the rest of the Samurai are charging on horses towards Colonel Bagley and the infantry in the final battle. Katsumoto is shown with a bullet hole in his shoulder armor, moments later he is shot in the exact spot. The bullet hole was there before he was shot.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
The seppuku ritual in the film is correct for a battlefield. A person committing seppuku on a battlefield would in fact have his head fully decapitated; later, more ceremonial rituals, involved leaving the head still partially attached. However, since General Hasegawa was committing suicide on a battlefield, his second (Katsumoto) would sever the head completely.