So... do you just boil the wine, or what?

Dear Tricia, I have no idea how I'm supposed to cook with wine.  I don't know what to cook in it, when to add it, or what kind to use.  Please help this clueless reader.

No need to be embarrassed of your clueless-ness here.  I understand how daunting it can be to try something that seems oh so gourmet, but its actually very easy and you have lots of options.  Let's talk fancy pants vocabulary first.  When you're talking about cooking with wine, what you're usually doing with it is deglazing the pan.  You know how after you cook meat or veggies in a pan and there's that layer of brown residue stuck to the bottom?   Well that brown stuff is where all the extra flavor is.  It's called fond and you want to get it off the pan and into some kind of sauce.  Pouring a small amount of liquid in the pan usually does the trick.  That's what deglazing is: pouring some kind of flavorful liquid in the pan to get the fond off the bottom.  You can use stock, you can use vinegar, or you can use wine.  Wine is great because it gets the fond off the bottom and adds tons of extra flavor at the same time. And it looks like you know what you're doing.
So let's pretend you're cooking a nice chicken breast in that pan.  And when you're done with the chicken, you take it out and set it aside and wonder, "Hmmmm, how can I get this delicious fond off the bottom of the pan and create a lovely little sauce to pour over the chicken and impress my girlfriend?  Oh, I know!  I'll use some white wine and it'll be so good that she'll be ready for make-out city later!"  So pour in about a quarter cup of wine (usually you'd want to use something you're ok with drinking, not that bottle in the back of your fridge that's been open since last Halloween)  and stir it around with a spoon or a whisk.  The little brown bits will be released from the pan and then you want to wait while the wine cooks down to a teeny tiny amount, which is called reducing.  After its reduced in half  (which is really a mere few spoonfuls), add some a touch of chicken stock and about a tablespoon of butter.  Whisk in the butter, throw in a bit of salt and pepper,  pour that over your chicken breast and you're done.  See?  You've learned the words reduce, deglaze and fond all in 2 paragraphs.  Watch as I expand your horizons even further.   Yeah.  Get ready.
Let's talk Italian food for a hot minute.   When I make marinara sauce (and I say that word like an American does, not like Giada on the Food Network with her overly-enunciated Italian accent, jarringly thrown in the middle of a sentence) I like to brown my onions, peppers and garlic and then get the fond (yeah, it comes from vegetables too) off the bottom of the pan with some red wine.  I pour in about a half a cup and let that reduce until its almost syrupy.  There's no rule here about how much to pour in.   You could pour in an entire bottle and let it reduce, it'll just take longer.  (And then what would you have to drink while you're cooking?)  After its reduced, I'll add the tomato sauce and then season it up.
 So let's go back and review this.  I know it sounds like a lot at first but it's super easy.  Add the wine (while the pan is still over high heat) and let it reduce by half.  Add a touch of stock, a tablespoon of butter, and season it.  
As far as what kind of wine to use with what, you usually want to use red with something heartier, or with red meat. White wine is great for fish and chicken.  Again, use something you'd want to drink.  Cooking with wine is not a way to get rid of bad or old wine.  Then your sauce will just taste like old wine sauce.  Eww.
When you get comfortable with sauces, start getting creative.  Put some chopped up artichoke hearts and a squeeze of lemon in white wine sauces.  Add some capers and fresh herbs to red wine sauces.  And always season and taste it.  Salt and pepper go a looooong way, especially if you're trying to get to make out city with your lady later. 
Let's also talk about open flames for a second.  You know how when you're watching chefs on TV and they pour some wine in the pan and it all catches on fire for a second?  Looks fun, right?  Well mostly likely that's not going to happen in your kitchen unless you tip the pan and try to invite the fire into the alcohol.  Also, wine doesn't have as much alcohol in it as cooking with sherry or hard liquor.  Those guys will flame up no problem, but if you're using a gas stove and are afraid of becoming a burn victim, just pour it in and don't tip the pan.