Because a toilet disposes of a relatively large volume of "solid" mass, it is important that the drain pipe leading from the toilet be equalized at current atmospheric pressure. This is accomplished using a vent pipe that is routed upward, typically ending 10-12" above the surface of the dwelling roof. "Solid" waste moves best in two modes - Traveling vertically, and traveling at a slope of 1-2" vertically per horizontal foot. Any slope greater than this will cause the "suspension media" (water) to travel around the "solid" waste.
Typically, in an American dwelling, there is a 1 1/2" to 2 1/2" vent pipe leading from every kitchen and bathroom in the dwelling. the pipe leads straight vertically to the roof and is protected on the roof with a lead "boot" that can be obtained from roofing suppliers.
In bathrooms, the vent pipe typically leads vertical from the toilet, because it disposes of the greatest ammount of solid waste. All other drains in the bathroom couple to the toilet vent u8sing "T" fittings that are attached at a level higher than any drain water could achieve. In the kitchen, where a nice window often frames the view of the kitchen sink, the pipe is sometimes offset horizontally, or joined by a "t" fitting to other kitchen drains (typically the dish washer).