Ink cartridges
A cartridge is a small sealed plastic capsule of ink. You push it into the pen's barrel, the nib section pierces the seal and ink flows. They are clean, easy and travel well. The downside: you are limited to the colours the manufacturer makes in their own cartridge format (Parker, Pilot and others use different formats).
Converters
A converter is a refillable cartridge. The same shape as a disposable cartridge, it fits into the pen the same way - but instead of a sealed capsule of ink, it has a piston or squeeze mechanism that draws ink from a bottle. A converter gives you access to the full range of bottled inks, which is where the interesting colours live.
Bottled ink
Bottled ink is cheaper per millilitre than cartridges, comes in hundreds of colours and lets your pen do what it was built for. A 50ml bottle of Quink outlasts a full drawer of cartridges. The process: dip the nib into the ink, turn the converter piston to draw ink in, wipe the nib, cap the pen.
Which to start with
Start with cartridges. They require nothing except the pen. When you have been using it for a few weeks and want to try a different colour, buy a converter and a bottle of ink. The First Fountain Pen kit includes a converter so you have both options from day one.