As you play and win more matches, you'll be rewarded with money to buy more guns and gears. From the Shop/Storage screen, you'll be able to equip your character, and buy new items with the money you've earned.
Selecting a weapon category (AR, SMG, Pistol, etc.) button will show you the particular mods that can be used with those weapons. Keep in mind that different weapons may not be modded in certain areas. Others may come pre-configured with built-in silencers or scopes, precluding the use of a mod. Refer to each weapon mod description by clicking on the mod, to see how to best customize your weapon to suit your play style!
Muzzle - An attachment to the barrel of the weapon
Scope - Usually a sight affecting attachment on the top of the weapon
Magazine - An attachment that affects the weapon's ammo capacity
Mods that you have purchased and attached can be detached and placed on another weapon. Please remember that each mod is restricted to a particular weapon class-for instance, you can't apply the extended magazine from a machine gun and use it to augment the magazine of a sniper rifle. However, you can apply that class-specific mod to any particular weapon within that class (you can take an Assault Rifle suppressor off your M4 and put it on your AK).