the weapons that light up

I was wondering how the weapons light (or glow) up, is this something to do with an advanced level crafting skill? I'd be interested to know

Its the primal weapons and Relic weapons you're seeing. The relic quest is a big one you'll do when you hit 50.

Some involve crafting but others are obtained outside from crafting. 3 types are the Relic line (Relic Zenith, Animus and Novus) and upgraded Soldiary weapons (i110 weapons), and upgraded Leviathan weapons

If you're familiar with the Ifrit, Titan, Garuda and Moogle Weapons, you can use Desynthesis to break these down and have a chance to get a Demimateria based off the primal (Demimateria of the Vortex from Garuda Weapons). You can use these and other mats to craft i95 versions of those weapons that have a "glow" based off their element. Please note this is a 3 star craft process.

There are also i90 crafted body pieces that have a small glow as well, which is also 3 star crafting

Ok, I may have some way to go then, thanks for letting me know.
The only primal fight I've been in, was a low level Ifrit battle, involving my character and three others

There are two sets of "Glowing" weapons.

The ones you're far more likely to see are Relics - they're class-specific weapons with a long quest chain attached. After you complete the relic, there's additional quests to undertake to upgrade the relics, which give them added power and cause them to start glowing various colours. Because the Relic itself is fairly easy to complete, very powerful, and its first upgrade is very easy to do, a huge number of players at level 50 will have their Relic, so you'll see these glows very commonly.

Most recently are the "upgraded" Primal weapons, which are extremely difficult to obtain (Four of them require some incredibly rare ingrediants to craft, which have only become available as of the patch a week ago; the other requires a super rare drop from a very challenging trial to upgrade from another weapon) - these also have an elemental glow to match the weapon in question.

Well the low level Ifrit once you hit level 50 will be a hard mode version of him for your Relic weapon and will then be a 8 player instance rather then 4.
Relic weapon you get once you do all 3 Primals Hard Mode Versions mind you and for the glow that is what used to be called a +1 relic which is now Zenith (lvl90)
There is a continuation of upgrades to this weapon in the form of atma, Animus and now Novus.
Myself i do not have a Novus yet but Animus glows much more then Zenith did :)

The Trials of Ifrit/Garuda/Titan HM also drop weapons that glow.