Verizon was a strong force in making Android what it is today. They pushed it ahead in spite of Apple and the botched opportunities.
It was one phone in one country. The number of Motorola Droids sold were very few compared to Galaxy S phones. Motorola was what got the US finally into Android, but HTC was already there in the rest of the world. Samsung was the one that really made Android mainstream.
With that said, what made Samsung strong was several things:
1) US specific: They came out with the Galaxy S on ALL carriers. The same happened with the GS2 minus Verizon, but that's still pretty decent. The S3 was where they really increased their lead and it helped once again with all carriers aligned.
2) HTC launched phones on single carriers in the US a lot of times, and fragmented the global picture. I think had they pushed the Sensation across all 4 carriers, they would've done pretty well to fight off the GS2.
3) HTC had major fragmentation. Had they not done a mid-year refresh every year, they would've had a solid line to fight against Samsung.
4) In a time where Android was completely unpolished, they did some things right. Stock android was missing EVERYTHING. Samsung introduced quick toggles and a lot of other useful features. They had a hardware accelerated launcher and browser when Android was running like a slideshow.
5) The whole marketing thing is probably bigger in the US. I saw plenty of HTC ads all around Asia back in 2011.
6) Samsung used a unified product portfolio with the Galaxy name. It makes it very easy to follow, like the iPhone.
7) Good marketing and brand recognition really helped them. Despite the fact that I think the stock Galaxy S4 is shit with TouchWiz and what not, it's the only phone people recognize from the Android side. It's really between a "Galaxy" or an iPhone for 90% of those smartphone shoppers. That's pretty sad.
8) Timing. They stormed the market in force when it actually counted. They may not have the best products today, but they had the best back then. HTC took too long to realize their mistakes and correct themselves at the HTC One. Far too late.
In my opinion Motorola started out pretty well, then HTC did a good job with the Desire and possibly Sensation. Samsung had great hardware features like fast CPU, great camera, and I would've gone with them through the GS2. By the time the GS3 came out, there was little reason to go with Samsung, and to me that was the turning point where TouchWiz showed its trashiness and the Nexus phones started rising due to the awesomeness of ICS and Holo. By now, I think there's little reason to even get a Samsung phone other than for its hardware specs and sd card capabilities. If you're talking about out of the box user interface, I'd pick an HTC One any day.