A time ram was what happened if two TARDISes attempted to occupy exactly the same time and space. The resulting paradox would destroy both TARDISes and cause extreme spacio-temporal destruction in its wake. The fate for those caught in a time ram was what the Third Doctor considered tantamount to "extinction" and "utter annihilation". (TV: The Time Monster)
As the Master and Krasis departed for Atlantis, the Third Doctor made an attempt to materialise his TARDIS inside of The Master's TARDIS with precise control over the frequency and time setting on his own TARDIS. He feared an imprecise measurement would make him place the time setting too low and miss the Master's TARDIS entirely. The other danger was setting the time too high, which would cause a time ram. He explained the lethal repercussions to Jo Grant, warning her that she had boarded the TARDIS at her own risk on this trip, after she had insisted to come along with him. The Doctor successfully materialised his TARDIS inside of the Master's, but accidentally allowed the Master's TARDIS to end up inside of his as well, causing both TARDISes to be snared inside each other. This standoff was eventually broken by force.
When the Master trapped Jo inside his TARDIS as Atlantis fell apart while Kronos rampaged, the Doctor threatened to send the Master into time ram — even though it would kill them both and Jo — to prevent him escaping with the Crystal of Kronos, which would allow him to dominate space and time forever. The Master called his bluff, realising he wouldn't kill Jo. However, Jo used the controls herself to cause a time ram. They were all saved by Kronos, who brought them into its dimension. (TV: The Time Monster)
When Magnus Greel's time cabinet was being drawn out of its zygma beam between the years 5000 and 1872 to materialize in 1935, the Fourth Doctor and Romana I risked triggering a Time Ram to prevent the creation of an alternate timeline and ensure Greel arrived in the past to be defeated in London in 1889; by materializing the TARDIS where the Time Cabinet was trying to arrive, they forced the Cabinet to return to its original path as the location it was trying to arrive at was already occupied. (PROSE: The Shadow of Weng-Chiang)
The Sixth Doctor performed a time ram against the remains of Iris Wildthyme's TARDIS at the centre of a dimensional nexus. Local conditions limited the effects to merely eliminating some possibilities from reality, but the force was still enough to destroy the rest of Iris' TARDIS. (AUDIO: The Wormery)
The Profane Virus of Rassilon was a theoretical computer virus that would force a TARDIS to Time Ram itself mere nanoseconds in its own past; the Master infected TOMTIT with nanobots that would infect the Doctor's TARDIS with the virus, but Paul Kairos- the mortal identity of Kronos- was able to purge the nanovirus into the higher dimensions and allow the TARDIS to divert its past self. (PROSE: The Quantum Archangel)
The first eight Doctors time rammed their TARDISes against the Master's TARDIS in order to defeat him in a pocket dimension where he had framed them by using a conceptual bomb which was erasing the Doctor's timeline. (AUDIO: The Light at the End)
Other time-controlling species were also able to utilize time rams. The Daleks mercilessly used a time ram against a factory default TARDIS and the Eighth Doctor's TARDIS when its defenses were removed. (AUDIO: Tangled Web, X and the Daleks)