What was the Battle of Komaki and Nagakute?
Komaki and Nagakute was the only direct war between two of the three unifiers. Tokugawa Ieyasu won the significant engagement, Toyotomi Hideyoshi won the campaign, and both of them learned things about the other that shaped the next thirty years.
It is a useful corrective to the idea that Sengoku wars were decided by battles. This one was decided by a negotiated settlement with a third party.
Why did it happen?
After Battle of Shizugatake, Hideyoshi was moving from "guardian of the Oda heir" to "ruler in his own right". Oda Nobukatsu, Nobunaga's surviving second son, objected and asked Ieyasu for support.
Ieyasu had his own reason: an unchallenged Hideyoshi would eventually turn on him, and Nobukatsu's claim gave him a legitimate banner to fight under.
Forces and sides
Hideyoshi is usually credited with something over 100,000 men across the whole campaign, Ieyasu with roughly 30,000. The Tokugawa never intended to fight that army head-on.
What happened?
Ieyasu occupied and fortified Komaki hill, and Hideyoshi built a facing line. Neither attacked. The standoff lasted weeks.
To break it, a Toyotomi detachment was sent on a wide raid towards Ieyasu's home province of Mikawa. Ieyasu learned of it, moved out at night, and caught the raiding force strung out on the march near Nagakute. It was destroyed, and two senior Toyotomi commanders — Ikeda Tsuneoki and Mori Nagayoshi — were killed.
Hideyoshi brought his main army up. Ieyasu declined to fight it and withdrew to his fortified position. The famous moment attributed to this phase is Honda Tadakatsu holding a river crossing with a token force to cover the withdrawal.
The campaign then stalled for months. Hideyoshi ended it by negotiating directly with Oda Nobukatsu, who made peace separately. With his banner gone, Ieyasu had no cause left to fight for and stopped.
Turning point
Nagakute was the tactical turning point, but the campaign's real hinge was Hideyoshi's decision to go around Ieyasu politically instead of through him militarily.
Outcome
Ieyasu sent his son to Hideyoshi as an adopted heir — in practice a hostage — and formally submitted in 1586, after Hideyoshi had sent his own mother to the Tokugawa as a guarantee. The arrangement held for the rest of Hideyoshi's life.
Ieyasu joined the Odawara Campaign in 1590 and accepted transfer to the Kanto, where he built the base that won Battle of Sekigahara.
Why it mattered
Hideyoshi never fought Ieyasu again, and the way he ended this campaign — buying out the ally rather than beating the man — is a good illustration of how he actually operated.
For Ieyasu, the lesson was that he could not be beaten in the field by anyone willing to fight him on his terms, and that he could afford to wait. He waited fourteen years.