There are lots of ways to make fire without matches and other modern conveniences. Quite a few of them are eliminated if you are not an expert survivalist. The worst one of these and the best known is rubbing two sticks together. It takes a long time to get fire this way. Your hands get sore and your arms get tired and if you are not good at it, you won't get a fire going anyway.
A fire drill is quite a bit easier, but you have to know how to make it and practice and skill are necessary for this one as well. I will put up a link for it for those readers who are interested.
Fire drills and fire bows
More fire drills and bows
video of fire drill in use
I wrote a post on how to make your own waterproof matches. That is great for a short-term disaster, but eventually you run out of matches. The same is true of cigarette lighters. They run out of fuel. You need to know how to make fire for the long haul disasters, too.
There are videos on the internet on how to make fire using a battery. You get one with wires on the ends or hook some up and set fire with it to something. One video has you hook wires to a pencil and you set that on fire. Do not mess with this method unless you pay careful attention to the videos and have some idea what you are doing. Batteries can explode and are full of acid that is very dangerous.
Magnesium and flint and steel fire starters are pretty easy to find to put in your go bag. Magnesium will need to be replaced after a while. It is not something you could rely on for the rest of your life for fire. Magnesium and flint and steel require a little practice to learn to start a fire fast. You would hate being cold and wet and starting out with your fire starting experience.
Fire pistons are a little pricey, but they do not require as much practice as other methods and don't get used up like magnesium does. You can get the fire started with the fire piston easily, but keeping it going and transferring the spark to the tinder is the tricky part. This is a problem you will have with most fire starting methods, so practice could help a lot.
video of fire piston demo and description of making one
Lots of these videos mention using something to really encourage the fire to start. Tree fungus, punky wood from a rotten tree interior, vaseline, and dryer lint are common suggestions. Some fire pistons have a place inside them to store some of these and get your fire going good enough to light your kindling.
Ferrocerium fire starting rods are a less common way to start fires more easily than the two sticks method. They are becoming more common and therefore easy to find. People are beginning to use them for their barbeques and stuff like that. They have a little magnesium in them and throw off sparks to help you light a fire. Not all ferrocerium fire rods are alike. Some have more magnesium in them so the sparks they make last a little longer. This gives you more time to get tinder burning before the sparks go out. ferrocerium fire rods
This is not a totally exhaustive list of ways to start fires without matches or a lighter, but it covers a lot of the more common ways. If you really get to be interested in this, you can find a lot more out there on the internet and elsewhere. It should be enough to give you some idea of what to look for when you decide what to put in your go bag and maybe pocket survival kit.