It is very easy to get confused, because there are an awful lot of different lightsabers in the Star Wars series, and it's difficult to keep track of them all. Bear with me here...
Which lightsaber is which?
The blue lightsaber
The blue lightsaber that Rey gets is Luke's FIRST lightsaber.
It had previously been Anakin's SECOND lightsaber.
Obi-Wan picked it up after dismembering Anakin in Episode III.
He held onto it for years, and gave it to Luke in Episode IV.
Luke carried it until his hand was chopped off by Vader in Cloud City in Episode V.
The hand, still clutching the lightsaber, can be seen falling from Cloud City near the end of Episode V.
At some point, someone obviously recovered the lightsaber, although we don't know who found it or what the circumstances were. Some time later, the lightsaber somehow came into Maz Kanata's possession, and in The Force Awakens, Rey discovers it - and ultimately takes possession of it, at least for the time being.
Finn uses this lightsaber in his fight with the stormtrooper.
Both Finn and Rey use it in their fight with Kylo Ren.
The green lightsaber:
After Luke lost his first lightsaber, he needed to replace it (what good is a Jedi with no lightsaber?), so he built a new one; this lightsaber has a green blade.
Deleted scene from Episode VI, in which Luke constructs his second lightsaber
He uses this, his second lightsaber, in Episode VI.
As the previous answers have said, Luke's first and second lightsabers are not merely different colors; the hilts of the two lightsabers look nothing alike.
Luke's first lightsaber/Anakin's second lightsaber/the lightsaber Rey and Finn use/the blue lightsaber
Luke's second lightsaber/the green lightsaber
The reason the hilts are so different is because Luke's first lightsaber was made by Anakin, to Anakin's personal specifications, whereas Luke seems to have modeled his second lightsaber on Obi-Wan's third and final lightsaber.
Obi-Wan's third lightsaber
Obi-Wan using his third lightsaber in his final duel with Vader
Why are they different colors?
A lightsaber's color doesn't affect its performance, and is a reflection of the color of crystal used to build it. Anakin built his second lightsaber with a blue crystal, so the blade was blue; obviously, it remained blue after Luke started using it.
When Luke built his second lightsaber, he used a green kyber crystal, presumably because it was the only one available to him; in any case, there was no reason for him to not use a green crystal - as I said, the color doesn't alter the weapon's performance.
The only real significance behind lightsaber colors in the Star Wars movies is that red lightsabers are almost exclusively wielded by the bad guys. Most Jedi in the movies carry blue or green lightsabers, although Mace Windu had a purple lightsaber, and yellow lightsabers have been used by Jedi elsewhere.
