Psychological egoism says that you never really do anything for another and it is about what you can get out of it even if it is just feeling good or trying to have a good opinion of yourself.
If you solely or mainly do it for yourself you are classed as a egoist. If human nature is inherently egoistic then psychological egoism is true.
Critics say that some people are like you but not all. Atheists tend to say that religion makes more people like that than need be. Consider how Jesus promised new mothers and houses and material glory in Heaven if you ignore the blessings of this world to sacrifice them for him. This is not sacrifice at all but is mercenary. It is calculating and far beneath the dignity of correctly understood egoism.
Strong psychological egoism means you only act for another when you get a reward. The so-called weak version of the theory says you only act in the hope of a reward. I would add that you can act to get a reward and also in the hope of getting another one. In that sense they are not mutually exclusive.
Take the strong version.
There are two types of it that are not exclusive to each other.
One is that you are after something for yourself.
Two is that you are helping others because it suits you. You are after what suits. The difference is you may feed the dog to get praise from the neighbours or you may feed the dog for you feel like it and it suits you. You feel you honour yourself by doing this good thing. Doing what suits because it suits is a form of strong psychological egoism and the basic form. You can help x for it suits and because it suits OR you can help x because it suits you to get their inheritance. See the subtle difference?
Take the weak version.
Two types,
One you expect a reward. Positive.
Two the reward you expect is doing it to find it suits you. Negative.
You can say you only act for you expect some kind of reward and the truth is we all do. Critics object though that expecting a reward does not mean you are acting because of that expectation.
True - if it is a future reward. But there are rewards built into your action.
Even just being able to do something for another is a reward. Moving is like a gift. Seeing is its own reward when you are blind.
Weak psychological egoism is in fact strong when you think, "I want to reward myself by increasing my expectation of a reward."