Epictetus, a philosopher from Ancient Rome, was one of the most well-known Stoics. He wrote a short book called The Enchiridion (or "Handbook") that outlined how you should live your life to attain tranquility. Despite common beliefs, Stoicism isn't about suppressing emotions, but cultivating a positive outlook on the world by getting rid of negative emotions and desires, and, like Buddhism, not being ruffled by external events.
From Epictetus's The Enchiridion (125 AD):
"A man's master is he who is able to confer or remove whatever that man seeks or shuns. Whoever then would be free, let him wish nothing, let him decline nothing, which depends on others; else he must necessarily be a slave."
Stoics separate things in our lives into two categories: things within our power, and things that are not. Simple enough. What is in our power are things like our opinions, aims, desires, aversions; what's outside our power are our body, job, reputations, and, in general, the external world. Now, we may have some influence over things like our body or our job, but on the whole, it depends on external circumstances beyond our control.
Once we've made this distinction, we can then see what we should concern ourselves with. Obviously, we should only strive to change things that are within our power, because if something isn't in our power to change, what use is there worrying about it? What use is there in bemoaning that a thunderstorm began during your picnic? You have absolutely no control over thunderstorms--it will happen whether you groan about it or not. The Stoics advocate that we have no desires and aversions related to such events. Instead, we should take them as they come, and learn to enjoy them. We should modify our desires to conform to what is and not vainly try to stop things that are beyond our control. If we eliminate desires and hatred of such things in the world, we will be able to attain tranquility.
So we should focus on what we can change. Namely, things that are internal to us. This might lead to change in the external world, but in order to succeed, we must strive to make changes in ourselves rather than the world. Sometimes, it's just about changing our specific goals. We should want to cultivate our reason and imagination. We should want to be moral. We should want to be a good citizen. All these things depend on how we think and act, something that's completely within our power no matter what external circumstances we find ourselves in. It may not be easy, but it isn't beyond our control.
Now, things that are semi-within our power, such as our body or job, have to be given a bit more thought. Instead of making goals such as "I will get a degree in physics," which is external and largely beyond our control, we should change the goal to be internal, such as "I will learn physics and do my best in whatever challenges are put against me." That is something within our power, and is the kind of thing we should make goals about. If we make goals like this, we'll never be disappointed (assuming we follow them through by working as best we can), and so won't be ruffled by external circumstances such as the contingencies that go into getting a degree (the specific questions asked on a test, the whims of the instructors and markers...) that are out of your power. This makes you the sole source of your triumphs and woes, giving you the power to change your life and focus on things that matter.
So that's a bit of background. Looking at the quote again, the first sentence is simply a definition of what it means to be a master or a slave. I think this makes sense: we want to gain certain things and avoid certain things in our lives, and indeed, we live our lives for the sole purpose of gaining or avoiding things (I mean "things" in the most general sense, so this includes internal qualities such as knowledge, love, tranquility, etc.). So if someone can give us what we seek or wish to avoid, then they rule over us, and literally determine our lives. We are a slave to them. For example, if our purpose in life is to be happy, and if the only way to do this is to get an injection of a happiness serum, then whoever controlled the happiness serum controls you, because attaining happiness is all you seek (in this example).
So we shouldn't desire anything that depends on others, in other words, things that are out of our control. For example, the weather is totally out of your control, so you should never desire anything of it. You shouldn't, according to the Stoics, desire that it will be a nice day, or that you will avoid being out on a bad day. Likewise, your reputation is beyond your control. Do what you will, but people will form opinions about you based on their own ideas and prejudices. Instead, we should focus on the things we can change. This, the Stoics tell us, will lead to tranquility and happiness, and I certainly agree with them. We'll be able to appreciate the world more, be able to view the world with a new sort of wonder. By being free of judgments, we instead experience the world rather than trying to change things we can't. Indeed, Stoics are often very happy people, experiencing the world without jaded indifference or complaining about things never being the way they wanted (this is the same as Buddhists and Taoists).
Thus, we should internalize our desires so that we are our own master. This is really what Stoicism is about: not being ruled by other people, their opinions of us, and unconscious things like the weather, but taking charge of our own lives. In theory, Stoicism isn't complicated: it's principles are quite simple. But putting them into practice is where the challenge lies. As with most ancient philosophies, these aren't just theoretical ideas, but ways to live our lives. That's one of the main differences between philosophy in the present compared to the past. In ancient times, and largely throughout history, philosophy was more comprehensive: it wasn't a discipline you just studied at school and wrote papers about, but a way to live your life. As the historian Paul Veyne said, "To truly be a philosopher was to live out the sect's doctrine, conform one's conduct (and even one's attire) to it, and if need be, to die for it." That last part is just gold: to find something that you would stake your life upon and die for. Because when you follow a philosophy or way of life that you would rather be burned alive for than abandon, then you know you truly belong to it and have found true meaning in your life.