What makes a muppet a muppet?

Physical Appearance!
The puppets are often molded or carved out of various types of foam, and then covered with fleece, fur, or any other felt-like material. Muppets may represent humans, anthropomorphic animals, realistic animals, robots, anthropomorphic objects, extraterrestrial creatures, mythical beings or other unidentified, newly imagined creatures, monsters, or abstract characters.
Contrastingly to ventriloquist 'dummies/puppets', muppets can be animated with their arms and other features, rather than being only mobile in the face and head.
Muppets are typically made of softer materials and also presented as being independent from the puppeteer, who is usually not visible, i.e. hidden behind a set or outside of the camera frame. Using the camera frame as the 'stage' was an innovation of the Muppets. Previously on television, there would be be a stage hiding the performers, as if in a live presentation. Sometimes they are seen full-bodied. This is done by using invisible strings to move the characters' mouths and bodies; the voice being added later.
Muppets tend to develop, as writer Michael Davis put it, 'organically', meaning that the puppeteers take time, often up to a year, slowly developing their characters and voices. Muppets are also, as Davis said, 'test-driven, passed around from one Henson troupe member to another in the hope of finding the perfect human-Muppet match.'
When interacting with Muppets, children tended to act as though the Muppets were living creatures, even when they could see the puppeteers.
Operation!
The Muppeteer or Muppet master always holds the Muppet above his head or in front of his body, with one hand operating the head and mouth, and the other manipulating the hands and arms, either with two separate control rods or by 'wearing' the hands like gloves. One consequence of this design is that most of the Muppets are in fact left-handed because the puppeteers use their right hand to operate the head whilst using their left to operate the arm rod.
There are many other common designs and means of operation. Amongst the advanced Muppets, several Muppeteers may control a single character; the performer who controls the mouth usually provides the voice for the character. As technology has evolved, the Jim Henson team and other puppeteers have developed an enormous variety of means to operate Muppets for film and television, including the use of suspended rigs, internal motors, remote radio control, and computer enhanced and superimposed images.
Creative use of a mix of technologies has allowed for scenes in which Muppets appear to be riding a bicycle, rowing a boat, and even dancing on-stage with no puppeteer in sight.

Walter in The Muppets film