What decides whether a ring can be resized
The band style matters most. Plain and simple set bands are usually straightforward. Eternity bands, with stones all the way around, and some channel-set or tension-set designs are much harder, and sometimes the honest answer is that resizing isn't a good idea.
The stones matter too. Diamonds tolerate heat well, but emeralds, opals, pearls, and many treated stones don't. They may need to be protected, removed and reset, or the join made with a laser so heat stays away from them.
Finally, how far the size changes. One or two sizes is common. Big changes put more stress on the setting and can change how the stones sit.
How sizing up and sizing down work
To make a ring smaller, a small section of the band is removed and the ends are joined. To make it larger, a matching piece of metal is added, or a plain band can sometimes be stretched slightly.
Either way, the band is reshaped round and the join is finished so it blends in. Metal added to the band should match the ring's karat and color.
Before you send it
Know your target size. A jeweler can measure your finger, or you can compare against a ring that fits the same finger well. Tell your repairer about any earlier resizing, and send photos of the stones and the inside of the band, where the karat stamp usually is.