You need to find a Power Armor Station. If you exit your power armor in front of this station, or interact with the station while wearing power armor, then using the station will give you the option to modify and repair the constituent pieces of the armor.
Sanctuary has a Power Armor Station, as well as the Red Rocket station you reach early in the game.
There are several ways to get armour pieces and/or frames.
From Exploration
Many of the power armour frames you will find will have randomly spawned pieces on them, and many will be incomplete. According to Fallout wiki, random spawns will be based on level: T-45 power armor pieces at level 1-17, T-51 power armor pieces at 18-30, T-60 power armor pieces at 24, and X-01 power armor pieces at level 26 or more.
There are a number of maps online (IGN, Fallout4map.com) which show the location of all the power armour frames in the game (both random and pre-set) among other things.
From Enemies
You will also find enemies wearing power armour (many of whom are pre-set). These enemies will not drop an armour frame but you can take the pieces off their corpse.
As mentioned in another answer you can take the frame if you make them exit the armour by getting behind them and targetting the fusion core in combat (or reportedly pickpocketing the fusion core before combat). Though apparently if you make an NPC get out of it's frame it will count as "stealing" every time you use it.
From Vendors
If you join the Brotherhood of Steel you will be able to buy T-60 pieces and a frame from Proctor Teagan on The Prydwen (not sure if or how often he re-stocks, but he does restock).
Rowdy at Atom Cats garage will also sell individual pieces of T-45, T-51 and T-60 power armour and a frame (they also have some frames of T-60 standing around you could steal).
You can also buy an empty power armour frame from Commonwealth Weaponry in Diamond City and KL-E-0 in Goodneighbor (if you buy a frame it spawns on the nearby Power Armour service frame).
So Many Frames...
What to do with the spare frames is up to you, they appear to be indestructible, some people have suggested leaving them in lakes or something if you don't want to see them.
Repairing
Yes, broken pieces will be placed in your inventory, and can be repaired and placed back on a frame.
Best Answer
Yes it is possible. The broken armor pieces have to be in your inventory, then you need to go to a power armor station (while not wearing the power armor) and go to craft mode (by pressing A on Xbox One, X on PS4, and E on PC)
Then you can see all the pieces you have in your inventory and you should be able to repair them by pressing Y / Triangle / T, no matter if they're completely or partially damaged.