The blue lightsaber in The Force Awakens is Anakin's blue lightsaber and the lightsaber that Luke lost in The Empire Strikes Back.

It is quite different from Luke's newer green lightsaber.

We don't yet know how Anakin's lightsaber was found or retrieved.

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Because it was Anakin's lightsaber

If you look at the picture in the other answer, you can see that the one given to Rey matches the one given to Luke in Star Wars Episode IV: A New Hope.

To compare to Luke’s green lightsaber:

As you can see, it looks nothing like the one from Star Wars Episode IV: A New Hope or Star Wars Episode VII: The Force Awakens.

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.

That is Luke's lightsaber, as Obi-Wan gave him his father's lightsaber, which he took after the fight he had with Anakin. In Episode VI, he had a new lightsaber, which is green, as already been pointed out he needed another lightsaber. So the crystal he got was green instead of blue. Originally his lightsaber in Episode VI was supposed to be blue, as seen in the original trailer, but was changed as it's said it was difficult to see with the blue sky for Tatooine.

But either way, the first lightsaber Luke got was his. It was given to him. And also, if Anakin did actually die, I'm sure he'd want his son to have his lightsaber. It would be like a family heirloom.

So everybody here, unfortunately, is incorrect. If you compare the digital remake of the original films to the 35 millimeter film version, you will see that Luke's first lightsaber from Ben is more green than it is blue.

Check out this guy's side-by-side comparisons for an idea.

You could probably hedge and say "it's SORT of blue", but I think it's pretty clearly on the greener hue, myself. ...and you can obviously see that they blue'd it up for the digital remake.

Lucas said the only reason they made Luke's green in Return of the Jedi, was because the blue didn't show up well in the fight on Jabba's barge. So really, they both were initially intended to be blue.

There is a color called cryn which resembles what people are calling ice. It has been proven that the color change (green) although seen and explained in comics was attributed to blue not showing up well in the landscape (outside).