A tee connector looks like the capital letter "T." A pipe is fed into the the vertical part, and the top cross-line, or the horizontal part, splits the flow in two. This is a very common fitting, available in a variety of sizes. Many tees are combined in a system, depending on what facilities the plumbing designer want to pipe to. A tee connector is not used for drain or sewage systems, since they are designed for input, but not output flow.
A wye connector does not really look like a capital letter "Y," but is very similar. It looks more like a straight line, with a diagonal line drawn to it, much like a branch on a tree drawn on a trunk. This connector is mainly used in drainage systems. Flow is with the wye. The inputs are on top of the "Y," and the output is at the bottom. Drainage flow is very rarely split into two outputs, but rather split inputs are combined to one output. For example in a house, drains from the sinks, toilets and showers are combined into one main output, which feeds the sewer or septic system. Wyes are used to accomplish this.
A manifold is a combination of two words: "Many" and "fold." A manifold is a large pipe, with many small holes. A large pipe is fed into one side, and smaller pipes are attached to the smaller outlet holes. This fitting is used in many systems, such as irrigation or solar heating applications. The overall goal of the manifold is to take one input, and split it into many outputs. Manifolds are not normally used in home plumbing applications, but are quite common in industrial applications.
When you design the plumbing system, look at all the fixtures on the blueprints. Take the main input pipe, and lay out where all the tees and wyes should be. In many homes, 3/4-inch lines are ran to all the fixtures, and tee off with 1/2-inch lines to feed the faucets. Drains are usually 3 or 4-inch diameter, with 2-inch diameter pipe wyed into the main output line.