You will often see this listed on gear lists provided by different organisations. I think this is something of a misnomer, as your whole pack is your survival kit. What it usually refers to however is a collection of things to help you start fires, treat water, cut things, keep warm, and signal people searching for you. Therefor, if you have everything to do this (i.e. matches/lighter, water treatment, knife, sleeping bag and clothes and a torch) there is no need to go out and get a separate survival kit. You may see so called survival kits for sale in various outdoor stores, but they are usually filled with things you will never use, like fishing line, snare kits, and crappy wire saws. All of these things may have some use, but if you are going to be fishing, hunting or cutting wood bring something that will actually do the job properly. In general these kits are a waste of money and weight. Discussed below are some of the items you need to survive in the outdoors that are commonly though of as being part of a 'survival kit'. Note, water treatment has its own section.
Doesn't need to be flash, a simple pocket knife will suffice. If you are chopping lots of ingredients for a group dinner you might want something longer and sharper. However, you won't want something too big and heavy, as you won't use it unless you're into your hunting.
That you need to bring these is seemingly obvious, but you would be surprised how many beginner hikers forget them. You essentially have three practical choices: lighter, matches or a flint and steel. A lighter is what I prefer as it functions better in the wet, works instantly and you aren't left with lots of bits of burnt wood. Personally, I recommend carrying at least two methods as lighting a stove, campfire or fireplace is very important. It is good if the backup method is weather-resistant. So either a flint and steel or waterproof matches (both are a bit of a pain in the arse, hence why they are my backup).
Doesn't need to be flash, often they are built into a pack, though not always. The rationale for carrying one is that if you get lost, when searchers are looking for you, a whistle is a good way of attracting attention as it can be heard a long way off and doesn't take much effort to use. That last point is important, because shouting or screaming takes a lot of energy, most people can't maintain it for long.
First Aid Kit
This does not need to be substantial, though it is often helpful if you are going in a group to bring a more substantial 'group first aid kit'. I highly encourage you to go on a first aid course and get more expert recommendations. I am not really a first aid instructor, and there are many, many different opinions on what you should have in a first aid kit. My personal opinion is that you basically need to be able to treat common injuries incurred while hiking: blisters, splinters, insect bites (includes bee and wasp), small cuts, hypothermia, and dehydration. There are of course many injuries that can occur while hiking that are much more serious, such as breaks, strains and sprains, severe burns and the list goes on. Constructing a hiking first aid kit is a balance between reducing the weight and covering yourself for injuries you think are likely to occur. The problem with trying to cover every conceivable injury is that you will end up carrying a whole extra pack worth of things. You have to work out what injuries you commonly get, and what uncommon injuries you want to prepare yourself for. This is what I carry:
Plasters, blister specific plasters are helpful (cuts and blisters)
Needle (splinters and blisters)
Alcohol swabs (for sterilising)
Strapping Tape (for blisters on longer trips, doubles as a compression bandage)
Gauze pad. (for burns)
Antihistamines (for hayfever and insect bites).
Antiseptic cream (for wasp stings, in summer the places where I hike are plagued with wasps).
Electrolytes (on hot trips).
You may think this list is a little light, but first aid while hiking is predominantly about preventing injuries through good risk-management; the process of identifying risks and reducing their likelihood of occurring. For example, preventing sunburn by wearing clothes that cover your body, or by wearing sunscreen; preventing hypothermia by wearing enough clothes so you don't get too cold, but not so many that you sweat (which can be dangerous in cold weather, because as your sweat makes your clothes wet, and when you slow down they get cold and lose their ability to insulate). Prevention is the best first aid, so think about this as well when preparing your first aid kit.
Notebook + Pencil
Just generally useful, for all manner of things from playing games, writing down errant thoughts and in emergency situations. A waterproof notebook (yes, such a thing exists) is a good idea, they can be bought cheaply from most good outdoor stores.
Extremely useful if you are going to an hut where you don't know if there will be material to start a fire, or if you are going to a campsite where fires are permitted. There are many options for fire starters, vaseline coated cotton balls, bike-tyre inner, and actual fire-starters are all common options.
A quick and light method of providing extra insulation to yourself or an injured party. Carrying one is a bit of a no brainer, they are cheap, light and effective.