Why did Gandalf leave Thorin and company to enter Mirkwood alone in The Hobbit?

In addition to the plot-point reasons provided in the answer by the other educator here, there are literary reasons why Gandalf leaves the expedition. With Gandalf present, there will be no opportunities for Bilbo to mature as a hero. Gandalf is just too good at saving the day. Indeed, every time that Bilbo has been in serious trouble up to this point, Gandalf has swooped in to the rescue.

With Gandalf temporarily out of the picture, Bibo has his encounter with Gollum, finds the ring, battles giant spiders, hides among the Elves of Mirkwood, confronts Smaug, steals the Arkenstone, and brokers a peace between the inhabitants of Dale and Thorin. If Gandalf had been present during these escapades, it would have been difficult for Tolkien not to have had Gandalf use his powerful wizardry and gifts of persuasion to solve these situations. Bilbo would not have had the opportunity to grow as a heroic character and gain the confidence that defines him by the end of the story.

Gandalf was necessary in the beginning of the story in order to prod Bilbo along and provide him with guidance. With Gandalf nearby, Bilbo and his companions are relatively safe. However, Tolkien needed to get the wizard out of the way in order for the real adventure, and all the peril it involves, to take place. If you compare the character of Bilbo at the beginning of the story to how he is at the end, it is unlikely that this transformation would have taken place if Gandalf had been there to hold his hand through the whole adventure.

Why did Gandalf leave Thorin and company to enter Mirkwood alone in The Hobbit?

Gandalf leaves Thorin and company at the edge of Mirkwood.  All the dwarves and Bilbo are very upset and concerned about his imminent departure.  Gandalf, however, will not be persuaded, telling his friends:

"It is no use arguing.  I have, as I told you, some pressing business away south; and I am already late through bothering with you people" (125). 

Although Gandalf does not openly disclose his plans to Bilbo and the dwarves at the time, Tolkien later reveals the purpose of his business in Chapter 19, "The Last Stage."  When Bilbo and Gandalf at last return to Rivendell on their journey home, Bilbo hears Gandalf tell Elrond about his secretive mission.  Gandalf had been to the White Council, which is a powerful council of wizards and elves, including Saruman and Galadriel.  They had met to discuss the growing threat of the Necromancer (Sauron) in Dol Guldur, south of Mirkwood, and the Council decides to use their collective powers to drive him away.

Why is it necessary for Gandalf to leave in The Hobbit?

Gandalf leaves the company of the dwarves and Bilbo during chapter seven, just as they are about to enter Mirkwood.  Gandalf gives them the excuse that he has pressing business to attend.  Later, Tolkien reveals that Gandalf actually left to attend a meeting of the White Council, a meeting between Saruman, Glorfindel, Radagast, and Cirdan to discuss the Necromancer in Dol Guldur.  Suspicions run high that the necromancer might actually be the resurgence of Sauron, the deceiver. 

Gandalf's leaving the company of dwarves and Bilbo has a significant effect on the story line, forcing Bilbo to prove himself as a professional burglar, against the spiders in Mirkwood and then later to free the dwarves from the Elven king.

In The Hobbit by J. R. R. Tolkien, where did Gandalf go when he left the journey?

Gandalf left at a few points during story. I assume you mean the one of the greatest length, and that is as the dwarves and Bilbo entered Mirkwood.

Gandalf is very specific when he tells the group that they must not leave the path as the enter Mirkwood because of the great evil that has now spread through much of the forest. In the new to DVD movie version, this danger is made clearer by an added sequence that includes Gandalf's counterpart wizard, Radagast the Brown. Radagast had detected a great evil growing in the forest.

In The Hobbit text itself Tolkien tells us "It appeared that Gandalf had been to a great council of the white wizards, masters of lore and good magic; and that they had at last driven the Necromancer from his dark hold in the south of Mirkwood" (201).

I found additional information in Tolkien: The Illustrated Encyclopedia (1991) by David Day. According to that source,  Mirkwood forest used to be called "Greenwood the Great" until the darkness became so powerful there.

The source of that darkness was a being known as "the Necromancer of Dol Guldor." Gandalf later found this to be Sauron the Ring Lord.  At that time, many had believed Sauron had been defeated when the ring had been cut from his finger by Isildur, but because the one ring had not been destroyed, Sauron's spirit was able to rise once again. He built the fortress at Dol Guldor with the Ringwraiths and quietly built power there for almost 2000 years when he re-entered Mordor.

In The Hobbit, why does Gandalf leave the expedition?

It's an interesting question, because to get a more complete appreciation as to what's going on in The Hobbit, it might be helpful to look outside of The Hobbit, and more towards the appendixes of The Lord of the Rings.

Gandalf takes his leave of the company at the end of chapter 7, when they've come to the edge of Mirkwood, claiming he has business in the south. Later, Bilbo will learn more about Gandalf's departure, that he'd left to attend "a great council of the white wizards . . . and that they had at last driven the Necromancer from his dark hold in the south of Mirkwood" (The Hobbit, chapter 19).

However, The Hobbit provides a limited view into a much larger conflict. For Gandalf (and here, I'm drawing from the appendixes of The Lord of the Rings), the adventures of The Hobbit, served as only a part of the much larger struggle against Sauron (referred to as "the Necromancer" in The Hobbit), who had been building his strength in Mirkwood. Indeed, Sauron's presence was at the very heart of Gandalf's concerns about Smaug—he feared that, when Sauron at last moved against Middle Earth, he would use the dragon as a weapon.

In The Hobbit, why does Gandalf leave the expedition?

I assume you are referring to Chapter 2 when the dwarves suddenly notice that Gandalf is no longer with them before they get captured by the trolls. Although at first we are told they are most put out by his unexpected absence - for he vanishes without saying that he is leaving - it all works out for the best, as Gandalf is able to return and to save them from the trolls who have captured them and are imminently about to eat roast dwarf for dinner. When Gandalf has tricked the trolls into staying out in sunlight, he explains why he left them:

"I went on to spy out our road. It will soon become dangerous and difficult. Also I was anxious about replenishing our small stoc of provisions. I had not gone very far, however, when I met a couple of friends of mine from Rivendell."

It is these elvish friends from Rivendell that inform Gandalf about the two trolls, and he fears that the expedition might have become "waylaid" by the trolls, so he hurries back to save them.