Christopher Steele and the Russian "pee-pee" Dossier
A constant through all Clinton campaigns seems to be dirty tricks and intended to control the narrative: leaking salacious claims about the opposition to distract from all the negatives on their side of the aisle (then lying about it, repeatedly when caught). It didn't have to be true, it just had to keep people talking and questioning things until after the election. Just like the Clinton Administration's involvement in sensationalizing the Obama is from Kenya or a Muslim and being the impetus behind the Birth of Birthers movement. In this case, the Clinton campaign was trying to do the same with the idea that Russians had gained compromising information about Trump (also known as "kompromat,") and had been "cultivating, supporting and assisting" him for an elongated period of time. They wanted the public to believe that Trump was the Manchurian candidate.
There's some nuances in the funding that the Democrat media uses to try to cloud up the issue, and the media got it wrong (FakeNews) by repeating the DNC talking points and claiming Republicans funded the Dossier.
The Washington Free Beacon (and their #nevertrump patron Paul Singer) hired Fusion GPS for dirt on Trump first... but the Free Beacon dropped the investigation once Trump became the presumptive nominee (May 3rd, 2016), and their funding had nothing to do with Steele and the dossier. 
The AP, CNN, NYT, MSNBC and other outlets magnified this by getting the story wrong (as usual), and misreported that the Dossier was initially funded by the conservative Washington Free Beacon, they later had to offer a correction -- and they tried to change the subject back to the fake Russia narrative. Also after Hillary had gloriously lost the election, her campaign and the DNC stopped funding the dossier (which had been almost completely compiled by this time), so the co-founder of Fusion GPS (Glenn R. Simpson) paid for last couple weeks of work, and then handed the dossier to British and American intelligence services to use to undermine the President and the election. While agents openly mocked the shoddy report, the FBI/Obama administration still used it to get FISA warrants to spy on the Trump campaign during an election, and the FBI got caught omitting the origins of the report to the judge. Misleading a judge is a federal crime and is being investigated as such. To date, no one has gone after Glenn R. Simpson for his culpability in this smear campaign, nor will the media blame the DNC or Hillary campaign for their part in trying to undermine the office of the President.
According to Comey, Trump "told [Comey] to consider having the FBI investigate the prostitutes allegation to 'prove it was a lie'".
It seems highly unlikely that the President would request the director of the FBI to investigate something to exonerate him, if he was guilty and there might be evidence the FBI could uncover.
Think about the bigger picture, here's a President that got caught saying/doing all sort of foul things, and he just waved them off. The man tweets from the oval office like a Tourette's victim screaming obscenities in a library, and they think he's worried about what the Russians might say he did? Trump bragged he could shoot someone, and not lose votes. Trump doing a few hookers in a Hotel, and paying them to pee on Obama's bed, is only going to help Trump with his base. So not only didn't anything in the Dossier seem troublesome, let alone worthy of compromising him, but it seems to miss the basics of who Trump is. That Dossier could only appeal to biased Democrats who refuse to think this through at all, and wouldn't vote for him if he cured world hunger, ended disease and caused world peace in the same weekend.