One problem most novices encounter is torn grain due to a combination of lack of finesse with the technique and tools not sharp enough. The latter is the easiest to start with - make sure your tools are absolutely sharp.
As for technique, assuming you are using a gouge, present the tool with bevel rubbing and positioned to make a slicing/shearing cut which will result in fibres being cut more cleanly. If you are using a scraper or carbide, switch to a gouge!
If you are still suffering with tearout, look at the piece of wood. The area where you are getting torn grain is the end grain (which is usually where it happens) and you will see two patches of tearout on opposite sides of the bowl. If your piece of wood has some softer areas due to spalting and it looks like this one might, try soaking the wood with either wood hardener or sanding sealer which will help to stiffen the wood fibres and make it cut cleaner once dry. If you do this, it won't take many cuts to get through the hardened section so you may have to reapply as you continue to turn the bowl.
