I'm sure some others will chime in here but I'll give a start.
Tannins do indeed come from the grape juice being in contact with the skins, seeds, stems, and aging in barrel. They dry the mouth and coat the inside of your cheeks, your tongue, and stick to your teeth. They are naturally attracted to protein so the effect of tannins will be reduced when enjoying the wine with meat or cheese. They also help in giving red wine structure, balance, and a certain degree of tannins are required to allow the wine to bottle age. The degree of tannin can be felt in the mouth as woody or chalky texture. Some find it desirable and some detest it. In my mind it's all about balance.
Dryness is purely a comment about the perceived sugar content in a wine. A dry wine has been fermented to the point that there is no sugar left. In some cases (ie. german riesling) fermentation will be stopped before the wine is dry and it is then "off-dry" and retains residual sugar. The vast majority of red table wine is dry (though is could be argued that some retain a small amount of residual sugar and the are a touch off dry). Last time I had Yellow Tail and Sutter Home both based a bit sweet to me and I believe they both retain small amounts of sugar (or add some after fermentation) in their wines to create a sweet sensation to appeal to the masses, but are essentially dry when compared to something like German Riesling, Port, or desert wines.
Hope that helps!