Counted cross stitch has few rules. The main one is to enjoy yourself. You may follow or ignore any of the tips listed in this FAQ and still be a "real" cross stitcher.
Generally, people first learn to do counted cross stitch on Aida and learn to stitch on linen or other evenweaves as they become more experienced. Judging by comments in this newsgroup, most stitchers who know how to work on linen prefer it to Aida. As always though, this is a matter of personal choice. Some very experienced stitchers prefer Aida.
An evenweave is any fabric which has the same number of threads per inch in both the vertical and horizontal directions. The individual threads might not all be the same thickness--you can see this in linen--but the number of threads is the same.
First, the traditional rule--stitch on Aida using a hoop and stitch on linen "in the hand". In actual practice, people do whatever works best for them. See section "6. Hoop or Hand?" for a discussion of the "in-the-hand vs. in-a-hoop" debate. See section "30.2 Hoops, Scroll Bars and Such" for more information on the equipment itself.
Most evenweaves aren't as stiff as most Aida. This can be a plus or minus, depending on your own preferences. The difference in stiffness isn't usually a factor if the fabric is worked in a hoop or in scroll bars.
Aida is worked with one X per square, while linen and other evenweaves are generally worked over two threads. This means that a 28 count (28 threads per inch) linen produces the same size picture as a 14 count (14 squares per inch) Aida. See section "18. Stitching On Linens and Other Evenweaves" for a more detailed explanation of stitching "over two."
Fractional stitches (1/4 stitches and 3/4 stitches ) can be much easier to do on an evenweave material. On Aida,the needle needs to punch through the middle of the little square in order to complete the stitch. This can made somewhat easier by using a small sized needle (#26 or #28). No "punching through" is needed on an evenweave, as the needle simply goes between the two threads. See section "12. Fractional Stitches" for a more detailed explanation of fractional stitches.
Some people find it easier to see the holes on linen and other evenweaves, others find the Aida easier.
The look of the cloth in the background is also important when selecting a fabric. Both texture and colour should be considered.
Aida is generally less expensive. Whatever fabric you choose to work on, always buy the best quality you can afford. The amount of time invested in a project can be quite large and is far more valuable than a small savings up front.
Also make sure to know the fibre content and if the fabric requires any special care.