There's a biography of Trump on the A&E channel that consists of a lot of interviews, including one with Roger Stone. It seems that there are three theories as to why Russia might have "Kompromat" (compromising material) on Trump:
1. Deutsche Bank (DB) was the only big bank willing to loan money to Trump after his companies' bankruptcies. DB is also one of the primary banks that Russian oligarchs have used to launder their money. They've paid something like $600 million in fines for laundering Russian money. Trump owed DB's real estate division $645 million and was not going to be able to make a $40 million payment, which would have put him into bankruptcy. Then, lo and behold, DB's Private Bank, a separate division of DB, loans the Donald $200 million or $300 million, and he's able to pay enough on the other loans to stay in business. The theory is that Russian oligarchs or mobsters or the Russian government somehow guaranteed the loan and have managed to maintain leverage over him. This seems pretty unlikely to me. It's more likely that DB's internal controls or corporate good judgement are fucked up.
2. The second theory is from the Steele dossier, involving the two women that Trump is alleged to have banged and who urinated in his bed, which was the same bed Obama slept in years before.
Trump owned the rights to the Miss Universe Pageant. In 2013 it was held in Moscow. Trump was excited about this, not for the profit potential of the pageant, but because it offered him an opportunity to improve his "brand" and his contacts in Russia. At that time, Russia was booming, and real estate was very expensive. He was trying to get a Trump Tower off the ground in Moscow, which would have been a huge project. Trump also had high hopes of meeting Vladimir Putin and building a relationship with him.
Trump's hotel room, according to one person interviewed who stayed in the same hotel for several years with his family, undoubtedly had cameras. The interviewee said he left with his family for a few days, and when he came back, the FSB (KGB successor) had left a copy of a Russian sex manual by the bed, opened to a page with advice for achieving orgasm. Apparently the poor guy had problems getting off, and the FSB knew it from watching him on camera.
Anyway, at 1:00 AM Trump comes back to the hotel room. Trump's bodyguard, Keith Schiller, was asked earlier in the day if Trump would like the company of 5 women. Schiller said that's not the way he rolls, and that was that.
Schiller was dismissed by Trump, who was up bright and early the next morning for all day meetings and the pageant. Looking at videos of Trump on that day, he's bright eyed and bushy tailed. I don't see how a 67 year old man could possibly have entertained two women that evening after 1:00 AM and still functioned the next day. And the part about Trump asking them to urinate on the bed because Obama was there is too strange to be true. I don't believe this theory either.
3. A third theory is that Trump could be blackmailed by the Russians because he collaborated with them on the 2016 election. That's been discussed here ad nauseam. Personally I'm not a believer in the collusion theory.
That leaves the schoolboy crush theory. I think there's something to that, that Trump does indeed admire Vladimir Putin. He was disappointed that Putin didn't show up for the Miss Universe pageant. He proposed to a couple of people working on the pageant that they go ahead and say Putin did show up, as people wouldn't know any different.
My own theory is that Trump's attraction to Putin during this period was motivated more by potential commercial opportunities than by admiration. Friends of Vlad, including a number of Oligarchs, have become fabulously wealthy. Undoubtedly Trump would have loved to extend his brand and empire to Russia, and a relationship with Vladimir Putin might have guaranteed that.
The Russian invasion of Eastern Ukraine and the Crimea put an end to Trump's Russian ambitions, at least for the time being, as sanctions imposed on Russia by Obama made that impossible.
How about since the election? Actually, I wish relations with Russia were a lot better than they are now. Trump is right when he says he's been hard on Russia, although he was forced to by Congress, by public opinion, and to try to counter the perception that Russia aided in his election. Trump probably does actually admire Putin. I don't think he really admires, for example, Kim. That's an act, part of his negotiating strategy to try to bring North Korea back into the civilized world. But there is something special between the Donald and Vlad.