That's going to differ from school to school, county to county, state to state. In general though "Honors" courses will be tougher and faster paced, but generally are better if you can pass, often weighting your GPA (that is, if you get a B it will still count as a 4.0 instead of a 3.0).
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algebra 2 is more advanced
Pre-algebra is where you just learn the basics of Algebra and Algebra two is way more advanced with new information and taking the concepts you learned in pre-algebra and algebra to the next level.
advanced algebra is hard simple algabra is easy
Algebra is a form (or subset) of mathematics.
algebra 1a is the first part of algebra 1 and algebra 1b is the second part. :)