Solo Mode is designed for a single person. You are given an droid that will tank and heal, leaving it up to you to deal most of the damage (though it can finish some fights on its own).
Story Mode is the basic group mode for a Flashpoint and typically requires a tank, a healer, and two DPS. Many Story Modes were low-level, with Hard Modes taking the same format but at a higher level and adding additional mechanics or difficulty, but since Patch 4.0 all SMs are now level-synced. There's also Tactical mode, which is designed for 4 players independent of role.
For the list of available Flashpoints and their available modes, see How does hard mode work? All SMs are soloable depending your capability. Some extras may not be doable — for example, the bonus boss in The Esseles requires 2 people to trigger.
There's also Operations (with Story and Hard modes), which are basically the same as Flashpoints except longer, harder, and requiring 8 or 16 people. (There are also "Lair Bosses" that are single-boss Operations — not to be confused with World Bosses which are not instanced.)
Additional fun fact: During the 3.x patch series, the Explosive Conflict operation was soloable on 8-man Story Mode. Thanks to Level Sync that is no longer the case.
Aurora Cannon is a group Heroic, not a Flashpoint. Heroics are normal side quests that are usually very difficult to do alone, rather than requiring multiple people; Aurora Cannon was one of the few exceptions, but Patch 4.0 made it so that it can be soloed.