Double running refers to having another person help run the flag. This can happen in one of two ways: two separate people running separate flags (dual running), or two people running the same flag in relay (relay running).
Relay Running is when two people run the same flag, with each person running the flag part of the way. Ordinarily, the person who carries the flag on the second "leg" of the relay race (into the stand itself) is a high armor character, such as a warrior or ranger. This person is generally short term conscript who has a speed boost or other defensive skills to help force a flag in through the other team. While the high armor conscript forces the flag in, the real runner typically goes back for another flag so they can bring it right out after the other team caps over the first flag.
The advantage in running a flag this way is as your runner approaches the stand (and takes the most damage) you have a high armor character. This reduces overall damage and decreases the chance of a spike death. This makes this tactic popular if your runner is a softy (such as an ele) or has DP. This tactic cuts the total time of running flags in half while also reducing the damage taken and risk of death as your flagger pushes in. Disadvantages of relay running include the temporary loss of the pressure from the high armor warrior, the risk of someone from the other team slipping away to return the flag as it is relayed, and the general tendency of teams that relay run to make positioning errors (giving up ground and/or overextending).
Teams typically relay run defensively; that is, they relay run when they need to quickly overcome a flagging disadvantage or when they need to force a flag in and then quickly cover getting capped over. A common example of this tactic is right after your team wipes and loses the flagstand, but the other team has not yet gotten a boost. If there is time, the wiped team will res up, quickly force a flag in, then relay run in a second flag to cover getting capped over. The purpose of this is to prevent a boost then quickly gain the stand back after the other team caps over you. In other words, you hope to quickly shift the tables by capturing and recapturing, causing the other team to send their flag runner away just as your runner arrives and caps, giving you a temporary 8v7 advantage and control of the stand. You then try to leverage your numbers advantage into a wipe. However, pushing in a flag through ice and a full 8 person team can severely drain your monks.
Relay running is a common counter to dual running.
Dual Running refers to two separate people running two separate flags. Each runner will cap, fight for a few seconds (typically unloading most or all of their energy in a short burst), then run back for the next flag as they regen resources.
The advantage of running a flag this way is you can maximize the damage pressure and time pressure on another team. If your dual runners can get flags to the stand in half the time as a single runner, within a cap or two the other runner will be severely behind. This time gap means the other runner will have less and less time to force in the flag before you get a boost. This "pressure" in turn invites your team to focus damage and snares on the other runner as he pushes in, burning the other teams monks' energy. Your snares and pressure further increase the time disparity on flags. As the runner has to spend more and more time running and less time doing anything else (such as spamming heal party, defending the base against gankers, or snaring your offense) he becomes ineffective and a drain on the monks. Your goal is to eventually either kill the runner, burn out their monks, or disable their offense as they are forced to play defensively to counter your speed advantage in running combined with the pressure you are putting on their flagger.
Dual Running is an offensive tactic; your goal is to pressure the other team by compressing the time they have to get a flag in while still unloading as much offense as possible. A common example of dual running happens after a team clears out its base of gankers, then moves to power play the middle by having your anti-gankers run the next flag rather than sending your runner back. This temporarily gives your runner more time at the stand and when your anti-gankers show up, you should have numbers and a time advantage.