A paramecium is a unicellular organism belonging to the kingdom Protista, so it isn't exactly a plant or an animal. The organism can digest food, move through water by propelling itself with cilia, and reproduce. As one of the oldest organisms on earth, it has evolved and developed very simple methods of defense, genetic exchange, and mobility.
There are several different species of paramecia, but all are ciliate protozoa. This means they use cilia to swim through water, and their one complex cell, a eukaryote, conducts all of the organism's basic functions. It doesn't divide labor between different tissues or cells like an animal. Instead, each paramecium is capable of an aerobic exchange, similar to breathing, reproducing asexually, ingesting nutrients, and expelling waste.
Physical Appearance
With good vision, it is possible to just make out the speck of a paramecium, since they are about .02 inches (.5 mm) long. They're better viewed under a microscope, where a slipper or kidney shaped cell can be seen. Lining the outside of its membrane are tiny, beating hairs called cilia. Cilia move in conjunction, like a line of oars on a ship, to move the paramecium through liquid. The eukaryote is even smart enough to navigate around obstacles and towards food.
When a paramecium encounters food, it swivels to move the food into its gullet. The gullet is a small opening, like a mouth. It's lined with other cilia to help "swallow" the bits of organic or decaying matter it eats, such as other unicellular organisms or bacteria. The food will continue down the gullet to get stored in food vacuoles until the cell needs energy.
It may also be possible to make out other rounded structures, called organelles, that sort of function like an animal's organs. One such organelle is the contractile vacuoles. The paramecium must keep osmotic equilibrium, which means the water pressure outside its skin and inside its body needs to be equal at all times. Contractile vacuoles pass water from inside the cell to outside, and vice versa.
Reproduction
Under most circumstances, paramecia reproduce by splitting themselves down the middle and giving each new organism half of the organelles. This is called binary fission, and is a simple form of asexual reproduction. Occasionally, a paramecium will meet another and exchange genetic material during a kind of primitive sexual reproduction. The membranous skins combine to make one giant paramecium, at which point the tiny micronuclei that hold all of the genetic material switch around. When it divides into four smaller paramecia, they now have new combinations of DNA.