A wood fence is made up of a few different parts and it's helpful to know what they are. To help you understand the terms used in your fence quote we've created a list of some common fence terms and an explanation of what they mean.
Pickets are the main portion of the fence. Different fence styles will use different types of pickets. A standard nail-up fence can use dog-eared pickets or flat-top pickets. A picture frame fence will only use flat-top pickets.
Posts are the main support structures for the fence. Typically posts will be installed every eight feet in order to keep your fence from sagging. A standard fence is six feet tall, and posts will need to be eight feet tall because around two feet of that will be below ground. Standard post thickness is 4x4". Because posts come in contact with the ground they are pressure treated so they are more resistant against decay and moisture.
A kickboard is a piece of pressure-treated wood that is placed on the bottom of the fence and is meant to be in direct contact with the ground. Because it's pressure treated, it's more resistant against rot and pests, which will help the rest of your fence last longer. Kickboards are generally used with picture frame style fences, but it's also possible to have a kickboard on a nail-up fence too. More details about kickboards can be found here.
A lattice, or lattice topper, refers to a decorative wood structure that's placed on top of the fence. The most common style is the diagonal lattice, which has diagonally crossing slats of wood which creates diamond-shaped holes in between. Lattices are commonly one foot tall, but it's possible to have a lattice that's up to two feet tall. You can see additional lattice styles here.
Rails are used as additional support for a fence. They are typically 2x4" thick strips of wood that run horizontally along the fence. Picture frame fences typically have two rails running along the top and bottom of the pickets for support. Nail-up fences can have two rails, though some customers choose to have an additional third rail for extra support and aesthetic purposes.
Trim is used to hide the fasteners that hold the pickets to the rails. This consists of small 1x1" strips of wood that help improve the overall appearance of the fence.