There really was a Bronze Age conflict which probably inspired Homer's epic.

Did any of the people in the story exist? No evidence for that exists. Did events happen as described in the story? Probably not, it was probably very embellished (like historical fiction movies/books). But we know from Hittite archives that there was a conflict between the city of Wilusa (the Greek name for Troy is Ilion, which is a cognate/loan word from the Hittite name for the city, Wilusa) and a people group called the Ahhiyawa (often identified as the Achaeans). The specific text refers to something called "the Wilusa episode" which involved hostility on the part of the Ahhiyawa toward Wilusa, which took place around the 1300s-1200s BC. All ancient textual indications and current scholarly consensus is that the Ahhiyawa are from the area of Greece/Mycenae.

The identification of Wilusa as Troy is also pretty sound, historically and archaeologically. All textual references that locate the city geographically put it in the area that the mound of Hisarlik sits, and there are mentions of such things as a water tunnel in Hittite texts, which have been found in excavation at Troy. So it's pretty well-accepted that Troy=Wilusa=Hisarlik.

So we know that there was a city called Troy, and a people called Achaeans, and that the Achaeans attacked the city of Troy. In addition, one of the kings of Wilusa mentioned in Hittite texts was Alaksandu, which is the Hittite version of Trojan prince Paris' birth name (Alexandros).

However, Homer wrote his epic centuries after it took place, between 1102 BC (Pseudo-Herodotus' date) and around 850 BC (Herodotus' estimation). We have no indication that it was intended to be an accurate retelling of events, or anything more than a fun imagining based around something that actually happened (like Inglourious Basterds and WWII, or "Cold Mountain" and the Civil War). All we know is that yes, there was a war, and the Achaeans attacked Troy, and that's about it.

For further reading, I suggest

Sagona, A G, and Paul E. Zimansky. Ancient Turkey. London: Routledge, 2009

Easton, D.F.; Hawkins, J.D.; Sherratt, A.G.; Sherratt, E.S. (2002). "Troy in Recent Perspective". Anatolian Studies (52): 75–109

Allen, Susan (July 1995). "'Finding the Walls of Troy': Frank Calvert, Excavator". American Journal of Archaeology 99 (3): 379–407

Wood, Michael (1985). In Search of the Trojan War. BBC Books; First Thus edition

The war in the way that Homer portrays it, as a clash of the supposedly most powerful forces at the time, a 10 year siege, and heroic almost godlike warriors, almost certainly did not happen. You have to make a distinction, Homer's Trojan War, did most likely not happen, rather I like to think of it as a "inspired by true events" kind of story.

If you instead asked "was there a war involving Trojans and Greeks", the answer becomes probably.

The location of Troy at Hisarlik, that Heinrich Schliemann designated in the 19th century, is mostly accepted as at least the inspiration, or maybe even location of a Trojan war. The mound at Hisarlik was occupied for a very long time. The first signs are from the 3rd millenia b.c, and the abandonment is dated to sometime in the early Byzantine period. This long period is split into 9 different levels, I-IX.

Level VI is most commonly accepted as Homeric Troy. Level VI started around 1750 b.c, and led a violent downfall in 1300-1250 b.c. This date roughly coincides with both Herodotus' and Duris of Samos' datings of the Trojan War.

At this time the Hittites ruled the Near East, and happily they have left behind many documents and letters. Some of these refer to a kingdom of Wilusa. Wilusa is thought to be the early, or simply Hittite, rendering of the name Illion/Illios, an earlier name for Troy/Troia. A king Alaksandu is also mentioned in a letter, a name that is very unusual in relation to other names of kings of western Anatolia at the time, even to a previous ruler of Wilusa, Kukunni. It may be assumed that Alaksandu is the Hittite rendering of an early Greek form of Alexandros, a name that one of the heroes of the Illiad, Paris, also goes by. So this is essentially Troy attested as a kingdom, and in written form, and these are some of the arguments that are made to connect Troy with level VI at Hisarlik.

But was there a war at Hisarlik? And how does Troy VI relate to Homeric Troy? In the Illiad, it is mentioned that Troy's walls are sloping, and that Patroclus cannot climb them. This is represented in Troy IV, which also has sloping walls. It should though be noted, that this is nothing special. Many other cities in the Near East had these walls, notably Hattusa, and even one of Troy's earlier levels, Troy II had them. Homer speaks of the wide straight roads that Hector chases through looking to say goodbye to his wife, and of a large gate where Hector kisses her goodbye, and where King Priam watches from the watchtower, Achilles slaughtering Trojan troops. Both of these features are represented at Troy VI. Specifically the very large, 3.3 meters wide southern gate, is often thought of as the place of Hectors goodbye to his wife.

A large problem with relating the mound at Hisarlik to Troy is the size. The area within the walls is only 200 meters in diameter. How could this possibly warrant such a massive invasion described by Homer? Many suggestions have been laid forward, but one particularly stands apart from the others, the idea of a lower city. Hisarlik lays in the middle of very arable land, and should have been able to provide for a much larger population of people than what is possible could have lived within the walls of Troy. The excavator at Hisarlik, Professor Korfmann found in 1988, with the help of magnometers, evidence of a lower city extending to the south and east of the walled settlement. Small ditches have been dug the entire way around, and Korfmann has suggested that these may have been initial lines of defence for the outer settlement.

Finally, I mentioned at the start that Troy VI came to a violent end at the start or middle of the 13th century b.c. Cracks in the fortifications suggest that this might have been an earthquake, but no other evidence for this scenario has been found. If we assume that it was not an earthquake that destroyed the city, then the next logical conclusion would be human action. Worryingly, there's is also not a great amount of evidence for this. Spearheads and other weapons have been found, along with mutilated bodies, pointing towards some kind of conflict, but not enough for a supposed 10 year siege.

The Myceneans, a culture originally from the Peloponnese was at this time looking to expand their holdings in the western Aegean, this is reflected in Miletus, a city previously under Hittite control under the name Milawata, and Lesbos, an island not very far from Troy.

The Hitties refer to a people in the west as the "Ahhiyawa", and in several letters the Hittite king calls the ruler of this land "my brother" suggesting that the kingdom of Ahhiyawa, must have, to the Hittites, seemed very powerful, as this designation was usually reserved for the Hittites powerful vassals in Syria, or the Pharaoh in Egypt. Because of the location given, the name of them, and the titles bestowed upon them, the people called Ahhiyawa are most likely the Myceneans.

So there was a modestly powerful people to the west of Anatolia looking to expand their holdings in the east. It is not entirely unthinkable that this may have brought them into conflict with Troy. This could potentially be the "real" Trojan war.

In conclusion, the mound at Hisarlik is most likely what Homer was telling stories of, although of course exaggerating a bit for entertainments purpose, and the Myceneans have most likely come into conflict with this city, which may have inspired the grand story of Hector and Achilles that we all know so well.

Here at the end I would like to add that I am no historian, I am simply someone interested in Troy, and I welcome any criticism of my post. Check out /u/kookingpot answer too, if you have not already, as it is very good, and much better sourced than my own.

Sources: The Trojans and Their Neighbors, Trevor Bryce.