Linking conjunctions のに, ように, ために, せいで, and おかげで

のに is related to ので, but it works in the opposite direction. While ので means "because of X, Y", のに means "despite X, Y". Like ので, this is more "objective" than other adversatives. のに also loves being attached to the end of sentences as an emphasis marker of disappointment or discontentment.

もう会えないと思ってたのに、また会うことができた。
Even though I thought I would never meet you again, we could meet again.

Just like ので, のに needs な after nouns and な adjectives

少し考えれば、当たり前のことなのに。
It's such an obvious thing if you think about it a bit.. (sounds disappointed)

ように essentially means "so as to", as in "walk so as to keep your back hidden". This treads a fine line between "for the purpose of" and "in such a way as".

足音を立てないように歩く
Walk in such a way as to not make noise.

If it helps, you can think of ように as a noun (よう = "manner") turned into adverb by に. The clause that qualifies よう becomes "adverbial" to the main verb.

ために literally uses the noun ため "purpose" plus に to make a conjunction.

それを確かめるために、彼は走っていた。  
He was running, in order to confirm that. 

In many situations, ために and ように are interchangeable with the meaning of “for the purpose of”, but not always.

ために can also be used to mean "for the sake of" rather than "for the purpose of". The nuance is different in English but in Japanese it is not always obvious. に can also be dropped.

あなたのためなら、なんでもするよ。
For you, I would do anything.

せい is a noun meaning "fault" used to indicate cause/blame/reason. It usually implies that the effect is undesirable.

お前のせいで俺はこんなことになったんだよ！
It is because of you that I became like this!

The opposite of せいで is おかげで, which implies a thankful feeling.

俺は、お前のおかげで救われたんだ。
Thanks to you, I was saved.