A woven fabric is made of interlaced threads: the warp threads are parallel to the selvage of the fabric (height wise). The weft threads are at right angles to the warp threads (lengthwise). The weft threads are woven through the warp threads : there are several methods of doing so, and they create different types of woven fabrics (see “weaving” in this glossary for more information).
You can use any fiber you like in a woven fabric, even fibers with elasticity. Note that, when no elastic fibers are used in a woven fabric, one can observe a slight elasticity in the direction of the weft threads, but never in the direction of the warp threads.
The grainline of a fabric can refer to one of these two directions, but most of the time, when making clothes, following the direction of the warp threads ( parallel to the selvage) is recommended, to avoid any distorsion of the fabric.
Woven fabrics have their own properties, and must not be confused with knitted fabrics, that are used for different purposes.