Solitaire
A solitaire holds one stone on a plain band. Nothing competes with the centre.
Because there is nowhere to hide, a solitaire asks more of the stone itself. It also has fewer small settings to look after than a halo or pavé.
Halo
A halo surrounds the centre stone with a ring of much smaller ones. The effect is to extend the bright outline around the centre stone; the centre stone itself stays the same size.
The trade-off is more small settings to look after over time.
Pavé
Pavé sets small stones along the band itself, close together, so the metal almost disappears beneath them.
It adds light along the finger rather than only at the centre. Worth asking about: how the stones are secured, and what happens if the ring later needs resizing, since a fully paved band is more complicated to alter.
Think about the hand, not the photograph
A photograph shows a setting from one angle; on the hand it is worn in three dimensions. A design that looks striking straight on may sit high on the finger and catch on clothing; a lower profile sits closer to the finger.
If the ring will be worn beside a wedding band, try to see how the two sit together. Some shapes leave a gap, which may or may not bother you.
Metal and upkeep
Metal choice affects the colour and weight of the piece, and the colour of the metal can influence how the stone's own colour looks. It also affects maintenance: some finishes need periodic attention to keep looking as they did at the start.
Any specific care, resizing or aftercare arrangement depends on the company you buy from and should be confirmed with them directly.