想起-To recall something, to have something come to mind, to remember something (after trying to recall it).
Example: 我倒底把錢包放到哪裏去了呢!啊!我想起來了。我放在餐卓上。(Where in the world did I put my wallet? Ah, I remember now. I put it on the dining table.)
The primary difference between 記住 and 記得 is one of emphasis. Both mean "to remember" something, in the sense of not forgetting to do something. The prior has a more pronounced sense of "to learn by heart" or to "learn a lesson."
So you would say 記住,人生短暫,不要讓往事毀了你一生。(Remember, life is short, let not the past destroy your life.)
On the other hand, 記得 is much more casual, as if you're asking someone to remember getting the milk during the grocery run, or to remind someone to do something. 冰箱的牛奶快喝光了,記得下班回家前在路上買一瓶牛奶喔。(The milk in the fridge is almost gone, remember (don't forget) to buy a bottle of milk on the way home from work.)
Or 記得明天是媽媽的生日…(Remember it's Mom's birthday tomorrow). If you say 記住明天是媽媽的生日, it would sound as if you have forgotten it for the last several years and the speaker is admonishing you not to fail to remember it this year.