As your adventuring through the levels you'll come across different types of core drops - Low Cores - Medium Cores - Upgrade Core (Highs) - Upgrade Core (Highest) and Force Cores. You can always tell if a core has dropped as it will make a 'bling' noise as it falls to the ground in Battle of the Immortals.

Cores basically add stats to your weapons and armors - low, med, high and highest upgrade cores add +'s to your equipment making them stronger or more defensive (stats go up to +7), whereas a force core adds a special attribute to slotted equipment like HP + 50.

Be careful when using cores on equipment which is +4 or higher,

if fail to upgrade to +6 your item will go back down to +3
if fail to upgrade to +7 your item will go back to Zero.

Low Core:

Low cores are used for increasing stats on low grade equipment, they cannot be used on equipment like Titanium so if you want to practice using a core, practice with this one.

Unequip your item your upgrading, right click on the Low Core and then left click on the item, if its successful it will tell you so and your item will now be +1.

Medium Core:

Medium cores are used for increasing stats on Titanium or lower graded equipment and are used the same way as a Low Core.

Upgrade Core High:

Red Cores are used for increasin stats on the high grade equipment and shouldnt be used on low grade weapon or armour. Osmium, Shadowtitanium etc, are what you need to be saving

and using Red Cores on, anything else is waste. Just like Low and Med Cores, right click to use.

Upgrade Core Highest:

The most rarest of the cores and white in appearance, can be used to upgrade Mythril grade equipment.

Force Core:

Unlike the other cores you can only use force cores on items which are slotted and you have to go to the 'Core Alchemist' to use them.

When speaking to the Alchemist, click on 'Upgrade' to upgrade your weapon and then a new menu will appear. The Item space is to put the weapon or armor inside you want upgrading, the core space is to put your force core inside - i also put inside a low core as it will increase its chance of being successful (note: you can put 2-3 force cores inside and can get a chance at getting a level 2 upgrade but its completely random what you get), the Catalyst space is where u have to put another slotted weapon / armor inside and this determines your special upgrade.