I've spent most of the discussions so far focusing on what the right approach is to philosophy, with only showing the wrong approach as a contrast. But in Objectivist ethics, there really needs to be some discussion of altruism because of its dominance and it's opposition to Objectivist ethics. It's a huge topic, so I only want to address some of the bigger features.
First, altruism is technically a philosophy that upholds other people as that standard of value. What should be immediately obvious is that if others are the standard of value, then their interests are more important than your own. This means that altruism is an ethics of self-sacrifice. Your own life is always put below the interests of others.
Altruism doesn't just occasionally come into conflict with your self-interest. It actually judges your moral worth by how much you sacrifice. See this article:
Altruism is impractical. It has no point.
There are many varieties of altruism, precisely because it has no point.
Sometimes in Objectivist discussion altruism is taken to be any system of self-sacrifice. Technically, it is the generic form that upholds other people in one way or another as being the ultimate value. There are other distinct forms of self-sacrificial ethics, including environmentalism which upholds nature in opposition to all men.
Altruism is the dominant ethics. In fact, many people can't imagine morality outside of society, the term is so tightly connected to other people. If you ask someone for morally praiseworthy actions, they'll list things like helping an old lady cross the street, jumping into fires to rescue people, etc. If you are stranded alone on an island, morality isn't seen as relevant.
Altruism is usually based on a particular kind of world-view. The view is that people are just not cut out for survival on this earth, and we all have to stick together in order to survive. Typical altruist morality discussions involve a drowning person, and whether you jump in after them. This is especially the case when contrasted with rational self-interest. So altruism is based on this doom and gloom view of the world, where everyone will die if we aren't all struggling together. The contrast between that and most people's actual lives is stark, but the comparison is rarely made.
Altruism is often put into a package-deal with benevolence. That means they're equated, or said to go hand in hand. This obscures the nature of altruism by providing it with a moral facade. Being nice to people you care about it what altruism hides behind. But altruism is about self-sacrifice. It's about valuing people for their mere existence, and raising them above your self. And as mentioned above, altruism says the most moral thing is the largest sacrifice.
Benevolence does not need altruism. There are plenty of selfish reasons for being benevolent. But by mixing the two, altruism hides its ugly nature while stealing the credit for something it has nothing to do with.
Altruism is often not a complete ethical system. It provides guidance in some limited environments, but in many cases it leaves you without a clue. Many activities are considered amoral.
And finally, many people try to argue that altruism is not incompatible with rational self-interest. This is flawed, and we have the tools to see why. When it comes to a standard of value, we can see there's a definite conflict. Either your own life is considered the primary value, and everything revolves around it, or the lives of other people are. The conflict means that they will naturally lead to different results.
What the people looking for compatibility are probably confused about is that our interests are not necessarily in disharmony with other people. Pursuing our own goals does not mean hurting others. Sometimes helping others is in our interest. This is true. But Objectivist ethics is not aimed at not helping others. The standard is our own life, and if that means helping others, then no problem. But we're not doing it because it helps them, but because it benefits us in some way. The standard must be our own life.
Keep in mind though that it all has to do with priorities. That's what ethics is about. We make decisions between our choices by picking the best. So the question of altruism vs. Objectivist ethics really boils down to the standards of value involved. Are we acting in order to help other people as our ultimate goal, or are we acting in order to help ourselves. This is why the two ethical systems are not compatible. There must be a choice between these, and whatever choice made decides which ethical system you're pursuing.