The transduction barrier was a quantum force field (AUDIO: Prisoners of Fate) that protected Gallifrey and the Time Lords from invasion.
These barriers could also be raised around Time Lord space stations, and TARDISes were unable to pass through them. (AUDIO: Trial of the Valeyard)
Breaches
The barrier was deactivated on the orders of the Fourth Doctor. This allowed the Vardans and Sontarans to invade Gallifrey. (TV: The Invasion of Time)
One TARDIS did break through, but had to jettison 9/10ths of its interior, and most of its circuits fused while its recall circuit burned away, leaving only the telepathic circuits functioning. (AUDIO: Prisoners of Fate)
A squad of Daleks which infiltrated Gallifrey lowered the transduction barrier, allowing them access to Gallifrey to invade. (AUDIO: The Apocalypse Element)
The Time Lord's vault on Tersurus was protected by an outdated transduction barrier which the Master was able to break through using Goth's TARDIS. Before the barrier could be updated and the break-in retroactively undone, the Master killed the vault operators. (AUDIO: UNIT: Dominion)
The Eighth Doctor breached the transduction barriers to land the TARDIS outside Madame President Romana II's presidential palace. (WC: Shada) The War Doctor did it again during the Last Great Time War to catch the Time Lords' attention about his return to his homeplanet. (PROSE: Engines of War)
Other information
Anti-Time was unable to pass through the transduction barriers. (AUDIO: Neverland)
Alternate timeline
In an alternate timeline, the Knights of Velyshaa breached the barrier and invaded Gallifrey. (AUDIO: The Sirens of Time)