What are the different types of solder used for?
A great question, and since a textbook could probably be written to answer it, there's probably not going to be any single answer. I want to provide a general answer tailored to hobbyists, and hope that people more knowledgeable can come in and tie up specifics.
Summary
Solder is basically metal wire with a "low" melting point, where low for our purposes means low enough to be melted with a soldering iron. For electronics, it is traditionally a mix of tin and lead. Tin has a lower melting point than Lead, so more Tin means a lower melting point. Most common lead-based solder you'll find at the gadget store will be 60Sn/40Pb (for 60% tin, 40% lead). There's some other minor variations you're likely to see, such as 63Sn/37Pb, but for general hobbyist purposes I have used 60/40 for years with no issue.
Science Content
Now, molten metal is a tricky beast, because it behaves a bit like water: Of particular interest is its surface tension. Molten metal will ball up if it doesn't find something to "stick" to. That's why solder masks work to keep jumpers from forming, and why you see surface-mount soldering tricks. In general, metal likes to stick to metal, but doesn't like to stick to oils or oxidized metals. By simply being exposed to air, our parts and boards start to oxidize, and through handling they get exposed to grime (such as oils from our skin). The solution to this is to clean the parts and boards first. That's where flux cores come in to solder. Flux cores melt at a lower temperature than the solder, and coat the area to be soldered. The flux cleans the surfaces, and if they're not too dirty the flux is sufficient to make a good strong solder joint (makes it "sticky" enough).
Flux Cores
There are two common types of flux cores: Acid and Rosin. Acid is for plumbing, and should NOT be used in electronics (it is likely to eat your components or boards). You do need to keep an eye out for that, but in general if it's in the electronics section of a gadget store it's good, if it's in the plumbing section of a home supply/home improvement store, it's bad. In general, for hobbyist use, as long as you keep your parts clean and don't let them sit around too long, a flux core isn't necessary. However, if you are looking for solder then you probably should pick up something with a rosin core. The only reason you wouldn't use a flux core solder as a hobbyist is if you knew exactly why you didn't need the flux in the first place, but again, if you have some solder without flux you can probably use it for hobbyist purposes without issue.
Lead Free
That's pretty much all a hobbyist needs to know, but it doesn't hurt to know about lead-free solder since things are going that way. The EU now requires pretty much all commercially-available electronics (with exceptions for the health and aerospace industries, as I recall) to use lead-free components, including solder. This is catching on, and while you can still find lead-based solder it can lead to confusion. The purpose of lead-free solder is exactly the same: It's an evolution in the product meant to be more environmentally friendly. The issue is that lead (which is used to reduce melting point of the solder) is very toxic, so now different metals are used instead which aren't as effective at controlling melting point. In general, you can use lead-free and lead-based solder interchangeably for hobbyist uses, but lead-free solder is a bit harder to work with because it doesn't flow as nicely or at as low a temperature as its lead-based equivalent. It's nothing that will stop you from successfully soldering something, and in general lead-free and lead-based solders are pretty interchangeable to the hobbyist.
Tutorials
There are plenty of soldering videos on YouTube, just plugging in "soldering" to the search should turn up plenty. NASA has some old instructional videos that are great, because they deal with a lot of through-hole components. Some of these are relevant because they discuss the techniques and how the solder types relate.
In general, if you got it at the electronics hobby shop, it's good to use for hobbyist purposes.
Lead vs Lead-free
60/40 lead solder melts at around 191°C (376 °F) (and is commonly worked with at around 300°C (570 °F)) and takes about 1.5 seconds to melt and form a bond, aka "wet". Good bonds are shiny and shaped like a "tent", not a ball. With just a little practice, you can get good or at least very competent using lead solder.
However, lead is a toxic heavy metal, so extended skin contact isn't good for you (and terrible for the environment if disposed of improperly). I like to wear very thin cotton gloves while soldering (but I don't always). Note that lead does not "evaporate" during soldering. The smoke you see is the flux. But you shouldn't be breathing flux either. I use a fan and filter when soldering. There is also the "breath out" technique to avoid the fumes, which is fine for small jobs.
Lead-free solder melts at 220 to 300°C (430 to 570 °F) (depending on the formula), and takes about 4 seconds to wet. Good bonds are not shiny, and it is harder to visually detect a bad joint, at least at first.
Simple answer: unless you are planning to sell the device you are soldering to someone in the EU, stick with lead-based solder. Lower soldering temperature and faster wetting time of lead-based solder means less chance to thermally damage your board and parts (and its cheaper). Electrically, you can use either. You can even use lead solder to rework a lead-free board. Of course, then it won't be RoHS.
Diameter of solder
Very thin solder, 0.020" (0.51 mm) dia or less, gives you a lot of control over how much solder you are putting down, and melts a bit faster. But you have to "feed" the solder into the joint at the right speed, and unrolling another foot from the roll every couple joints gets old. I sometimes don't get enough solder into the joint with thin solder because I don't/can't feed it fast enough. Good for hand-soldering fine SMT parts.
Thick solder, 0.050" (1.3 mm) dia or more, is good for making big joints, like heavy gauge wire or leads on a TO-220 regulator. But it is easy to put down too much solder, and seems to melt slower as the solder itself acts as a heat sink.
I generally prefer "mid sized" solder, 0.025 - 0.031" (0.64 - 0.78 mm) dia, for most work. It gives me a balance of controlling how much I put on a joint, without the hassle of feeding the hair-thin stuff.
Flux
Electronics flux can be rosin, water-based, or no-clean. All are about the same quality as far as de-oxidizing the copper so a good solder bond can be made.
Rosin flux leaves an ugly, sticky residue. Cleaning it off takes either a LOT of water, or a (nasty) chemical solvent. You shouldn't leave it on, as it is slightly corrosive, and can also be somewhat conductive. This is falling out of usage due to the environmental impact of cleaning.
Water-based (aka Resin) flux is less ugly, not sticky. The stuff I have used leaves a white film behind. I have heard this film can cause problems with long term reliability. Some people just leave it on, but removal takes only a moderate amount of water.
"No-clean" flux is resin flux that burns or boils off, leaving almost no residue.
"60/40 lead solder melts at 315C"
Rubbish. 60/40 Sn/Pb solder has a pasty range of a few degrees C and solidifies at 183 deg C. A eutectic solder such as 63/37 Sn/Pb melts and solidifies at 183 deg C.
"However, lead is a toxic heavy metal, so extended skin contact isn't good for you"
Again, not true. Lead cannot propagate through the skin. It can only be ingested by transference from fingers to food or cigarettes, or through inhalation if atomised.
"Lead-free solder melts at 340 to 370C"
An alloy such as 96.5Sn/3.0Ag/0.5Cu is liquidous a t 217 deg C.
"Electronics flux can be rosin, water-based, or no-clean. All are about the same quality as far as de-oxidizing the copper so a good solder bond can be made."
Wrong. Different fluxes have different levels of activity and will help wetting and coalescence to different degrees.
"No-clean" flux is resin flux that burns or boils off, leaving almost no residue."
These are commonly synthetic fluxes that do not "burn off' but remain after the joint has solidified and are inert so do not pose any long-term risk to the circuit from hygroscopic action or ionic contamination.