Types of automobile tires

Performance (& racing) tires
Performance tires tend to be designed for use at higher speeds. They often have a softer rubber compound for improved traction, especially on high speed cornering. The trade off of this softer rubber is a lower treadwear rating.
Performance street tires sometimes sacrifice wet weather handling by having shallower water channels to provide more actual rubber treat surface area for dry weather performance; the ability to provide both high levels of performance on both wet & dry pavement varies widely between manufacturer & even tire models of the same manufacturer, & is a highly active area for research & development, as well as marketing.
The ultimate variant of performance tire has no tread pattern at all & is called a slick tire. Slick tires are not legal for use on public roads in most jurisdictions due to their extremely poor wet weather characteristics, but cheater slicks which circumvent the literal wording of the law, if not the intent, are available.
R compound tires, similar to cheater slicks, are considered as street legal, but are in fact designed for racing, with minimal tread & ultrasoft rubber. They can typically be distinguished by very low treadwear rating. NOTE DOT does not "Approve" tires. The DOT symbol is only a mark that confirms the tire is "Certified" by their manufacturer as capable of meeting applicable standards.
The highest performance tires actually designed to be driven on the street are often called summer or three-season tires, since they are optimized for ultimate warm weather wet & dry performance at the expense of snow & ice traction; they therefore must be replaced with winter or all-season tires, if the vehicle is to be driven much in the winter.

Winter (snow and ice) tires
Winter tires are designed to provide improved performance under winter conditions compared to tires made for use in summer. The rubber compound used in the tread of the tire is usually softer than that used in tires for summer conditions, so providing better grip on ice & snow, but wearing more quickly at higher temperatures. Winter tires often have fine grooves & siping in the tread patterns that are designed to grip any unevenness on ice. Winter tires are usually removed for storage in the spring, because the rubber compound becomes too soft in warm weather resulting in a reduced tire life.
According to California Vehicle Code Section 558, "A 'Snow-tread tire' is a tire which has a relatively deep & aggressive tread pattern compared with conventional passenger tread pattern".
Dedicated winter tires will bear the "Mountain/Snowflake Pictograph" if designated as a winter/snow tire by the American Society for Testing & Materials. Winter tires will typically also carry the designation MS, M/S, M+S, M&S, or the words MUD & SNOW (but see All-season tires, below)
Many winter tires are designed to be studded for additional traction on icy roads. The studs also roughen the ice, so providing better friction between the ice & the soft rubber in winter tires. Use of studs is regulated in most countries, & even prohibited in some countries due to the increased road wear caused by studs.
Other winter tires rely on factors other than studding for traction on ice, e.g. highly porous or hydrophilic rubber which adheres to the wet film on the ice surface.
Some jurisdictions may from time to time require snow tires or traction aids (e.g. tire chains) on vehicles driven in certain areas during extreme conditions.
 

All-season tires
These are an attempt to make a tire that will be a compromise between a tire developed for use on dry & wet roads during summer, & a tire developed for use under winter conditions, when there is snow & ice on the road. However, the type of rubber & the tread pattern best suited for use under summer conditions cannot, for technical reasons, give good performance on snow & ice. The all-season tire is therefore a compromise, & is neither an excellent summer tire, nor an excellent winter tire. They have, however, become almost ubiquitous as original & replacement equipment on automobiles marketed in the United States, due to their convenience & their adequate performance in most situations.
All-Season tires are also marked for mud & snow (e.g. M+S, M&S, etc.) the same as winter tires. However, due to the compromise with performance during summer, winter performance is usually not comparable with a winter tire.
 
All-terrain tires
All-terrain tires are typically used on SUVs & light trucks. These tires often have stiffer sidewalls for greater resistance against puncture when traveling off-road, the tread pattern offers wider spacing than all-season tires to evacuate mud from the tread.
Within the all-terrain category, many of the tires available are designed primarily for on-road use, particularly all-terrain tires that are originally sold with the vehicle.
 

Mud tires
Mud terrain tires are characterized by large, chunky tread patterns designed to bite into muddy surfaces & provide grip. The large open design also allows mud to clear more quickly from between the lugs.
Mud terrain tires also tend to be wider than other tires, to spread the weight of the vehicle over a greater contact patch to prevent the vehicle from sinking too deep into the mud.
Depending on the composition & tread pattern, many mud terrain tires are not well suited to on-road use. They can be noisy at highway speeds, & due to the open tread design, they have less of a contact area with the road, limiting traction. The large lugs on mud tires tend to tear & chip on roads, because they are made from hard rubber compounds that do not bend easily.
Mud tires are also marked for mud & snow (e.g. M+S, M&S, etc.) the same as winter tires.