It doesn't happen often that I make more than a minor brewing mistake -- something like being late on a hop addition, or mashing in a little cold. Unfortunately, with my Roggenbock, I pitched at 64F with the intention of bringing the temperature down quickly. Instead, I forgot about that, put the beer behind the partitions, and finished off my work day -- and almost three subsequent days -- without thinking about the fact that it needed to be chill-fermented. By the time I got back to it, fermentation had started in earnest. Now I'm concerned about off flavors, but there's nothing to do about that except wait. Fortunately, I've read that WLP835 is pretty temperature-stable up to the low 60s; it also needs a diacetyl rest at the end anyway, so that should clear things up a bit at the back end.
Ultimately, there's so much that can go wrong in homebrewing, it's useful to know how to fix common problems, or recognize those that don't need fixing. Here are some process and production issues and how to deal with them, sorted by the part of the process you will experience them.
Stuck grinder -- Unload the hopper, work through the problem, and slowly reload while grinding. Usually the sticking is caused by a small stone (or other physical impediment) or a rusted cylinder. Obviously, physical blockages should just be removed. Rusted parts need to be taken off and cleaned/relubed individually. If it's a part you don't want to remove -- which is most of them on a typical grinder -- wrap a paper towel around the cylinder on the inside and spray some WD40 on the exterior, then run the grinder through slowly a few times. Remove the paper towel, and you should be good for another six months of grinding.
Unground barley -- This is especially problematic if you've run the grinder backward, which usually throws the malt down the outside of the cylinders and does not actually grind them. To separate the good from the bad and regrind only the bad, you can usually bang on the outside of the storage bucket until the ground stuff falls to the bottom. Skim as needed. Alternatively, you can do this process scoop-by-scoop in the grinder hopper, banging out the ground grain until all that's left is unground, then grinding that in the normal manner. I've never done the latter, preferring a few overground grains to a big pile of annoyance presented by the second.
Completely full mash tun -- I use a 15-gallon cooler as my mash tun, and a 20-gallon batch typically fills it most of the way up. If you're smart about it before you start mashing in, you can split the mash up into 2 segments, pouring in ~50% of your desired water on 1/3 of the grain (i.e. a very thin mash). This is easy to stir, and you can then add ~20% at a time of water, then corresponding percentage of grain, until you're done, stirring completely at each step. If you didn't think about it beforehand and are looking at most of your grain in the mash tun as well as only part of your water, it's a little harder to recover from. In this case, take large scoops out and put them in any spare container you have, removing up to half the volume of the tun; then add whatever water you can until the mash is sufficiently thin to stir. Now add back the grain stagewise, as though you planned it that way in the first place.
Low temperature -- This one is much easier than the next (high temperature). I always plan for a low temperature by mashing relatively thick; it's easier to add boiling water and warm the whole thing up than to do anything else. You can also treat this like a decoction mash: scoop out large volumes of mash and boil them. This may degrade the boiled portion, but that degradation is a small price to pay for the mash actually, you know, working out. Just be sure to do the calculations before any of this, lest you suffer from...
High temperature -- Ice cubes are easiest. Encased freezer packs also work, but many of these aren't heat-tolerant, so I caution against using them. I've used a couple other methods in the past, none of them particularly fun. First, drain your wort -- immediately. Let the outgoing liquid cool a few degrees below what you want the mash to be, then re-introduce it. If it hasn't been overheated for too long, most of the enzymes will be intact and you'll still get full action from your wort.
Stuck sparge -- Most annoying of all, and there are plenty of alternatives out there. I've had to empty my mash tun before, which is a process that just kills a regular brew day, but recovery is fortunately easier than it seems like it could be. Once you've unblocked your drain, you're ready to re-introduce the mash. First off, thin your mash! Empty the tun into buckets or other container. Next, add back a small volume along with some sparge water. You're going to have to continuous sparge for a little while (or you can just do the whole thing), so get ready for a loss of 45-60 minutes. Happily, the process isn't that bad, it just takes some babying. Add more mash, add more liquid, etc., until the mash is back in and flowing ok. If you're a batch sparger, do your normal 2 or 3 runnings, but stop the runnings at the volume you originally expected. It's going to be fine.
Long drain -- Usually the draining process takes 10-20 minutes at most if you're batch sparging, or up to an hour for the whole process if you're continuous sparging. If the drain takes a long time, it does not harm the beer, but it can still be frustrating. If you're suffering from a long drain on a batch sparge, consider doing a continuous sparge. You can do this midway through simply by adding liquid while the drain is going on. The thinner mash will drain a little quicker, and though the end result might not be significantly faster, it should save a little bit of time. Also note that you can continue to pour water into the mash as you approach the end of the draining process if it will keep the pressure up.
Slow chill -- It's imperative that you chill the wort quickly once it's done boiling, and there are really no good shortcuts. If you don't have enough ice, you're kind of boned unless the ground water is cold. With a counterflow chiller, you can get the cooling needed on 5 gallons in a matter of 20 minutes without problems if the incoming water is sufficiently cold. For counterflow chilling, note that the flow rate on each side should be similar to minimize waste water and offer significant cooling. (Handy tip: Use the hot waste water to wash up -- you'll finish cleaning much quicker that way.) If you don't have a counterflow chiller or a plate chiller or a coil chiller, note that cooling is all about surface area, conductivity, and temperature differential. You can wrap your plastic output tube around something metal and shove it into a bucket of ice water. I've used a copper pipe as the core and zip ties to hold the wound-up tube. Fancy hint: braid your zip tie between the tube turns to increase surface area.
Stuck chiller -- Depending on the size of the output, you can also get a stuck chiller from the protein precipitating out of the wort. I have no good solution to this except to blow out the tube in reverse and start again. The time it takes to chill is far longer if your chilling mechanism is plugged up than if you just stop and fix the issue to begin with.
High temperature -- High temperatures can be beer killers. Put your fermenter in a disposable turkey roasting pan (they're large and flexible), add ice or ice packs, and put an old shirt on the fermenter (as in, dress the bucket or carboy). Be sure the bottom of the shirt is exposed to the water, pour some water over the top of the shirt, and turn on a fan. Add ice or ice packs to the water once if not twice a day, and be sure to swap shirts every two days -- those things turn manky. You'll also need to replace the water (it turns manky too) or add iodine salts to ensure it stays sterile and doesn't start growing disgusting crap.
Low temperature -- In the winter, low temperatures are common. You can usually deal with this early on by wrapping the fermenter in a blanket, as the internal heat of fermentation will increase the pre-fermentation temperature by a couple degrees naturally; keeping this heat in can raise it by upwards of a dozen degrees (F) without effort. For a longer-term solution, or later in the fermentation period, and if you can't put the fermenter in a warmer place in the house (like if you don't want a fermenter sitting in your tiny New York City living room or something), give your fermenter a hot bath or shower. Your 120+ F water can easily increase the temperature in your fermenter. If you're more adventurous and trust your sanitizing methods, pour a gallon or more of the fermenting brew into a pot and, stirring constantly, raise its temperature by the amount you think you need. By stirring constantly you won't burn your beer and you'll get consistent heat distribution through the liquid. If, for instance, your beer is at 60F and you need to ferment at 68F, you can heat 1/4 of the liquid to 90F without a problem; the resultant beer temperature should be around 68F. Repeat if needed, but remember that every time you open the fermenter you have the potential for infection.
Out of bottles or caps -- Any bottle works, including wine bottles. Indeed, wine bottles are much easier to come by than old beer bottles that are homebrew-appropriate. If you get a wine bottle, use a small piece of wood to hammer the cork back in after you've filled it. Complete the seal by pouring wax around the rim and over as much of the top of the cork as possible. This will make the seal air-tight.
Beer down the shirt -- Don't spill beer!