Alternate mode
An alternate mode (commonly shortened to "altmode" or "alt mode", also known as a "trans-form" or a "transform") is a term used to describe a Transformer's non-humanoid form, whether it be vehicle, animal, or otherwise. Various, more specific, terms can and have been unofficially used. Vehicle mode, gun mode, tank mode, etc.
Some Transformers don't have a humanoid form, so in those special cases, the altmode is perceived to be the mode which is more geared towards utility or disguise while their robot mode is the one geared more towards interaction with other Transformers. For example, for the G1 characters Laserbeak and Ravage, their animal forms are their primary mode, and the cassettes their alternate modes. Even Grand Slam and Raindance, who transform from cassettes to vehicles, would be thought of as having a cassette alternate mode, as their vehicle forms -- while not humanoid or animalistic -- are at least self-mobile.
Transformers with more than two modes are usually considered to have multiple altmodes. (Punch/Counterpunch on the other hand has one altmode and two robot modes.)
In the cartoon episode "The Killing Jar," the Quintesson scientist refers to the Quintesson guard's transformation as his alternate mode, the first such use of this term in the official fiction.
The earliest recorded use in the fandom of the shortened term "alt mode" is an alt.toys.transformers post from March 1994 [1]. From the context, it appears that the term was already understood by fans at that time, as the post's author does not explain what he means.
Beast mode
"Beast mode" is a term used to describe a Transformer's alternate mode when the alternate mode is of the form of an animal. The term was first used officially in fiction relating to the Beast Era. When stranded on a planet with high levels of Energon radiation, Maximals and Predacons took on beast modes from the native life on that planet. The partly-organic nature of the beast mode protected them from the radiation.
In taking on such beast modes, the Transformer did more than simply mimic the life form, he also took on some characteristics of the animal. This affected the Transformer in various ways. The idea that beast mode flesh can be used for protection from Energon radiation was also used in the IDW continuity by the Dynobots. Spotlight: Shockwave
Since then, the term "beast mode" has been used to describe the alternate mode of any Transformer having an animal form, even if wholly mechanical and unconnected to energon radiation protection.

