Anyway, the "two notes" of the title are about:
1. Iverson bracket notation: writing say "[x ≥ 5]" to mean "1 if x ≥ 5, and 0 otherwise" (casting the boolean condition to a 0–1 integer, in programming terms). He has good examples of how it seems unnecessary or innocuous at first (one starts out writing "1" for an indicator function, or putting it in subscripts), but as you do more and more manipulations of sums it grows on you and you realize how useful the convention is. (IMO you can work through the book Concrete Mathematics, to experience this for yourself.) Incidentally, using square brackets seems to be something Knuth introduced; Iverson himself had used regular parentheses.
1a. There's a historical sidenote on the expression 0^0, with an impassioned defence of defining it to be 1. (It's undefined as a limiting form, but for the value itself it makes a lot of sense to define it as 1. Some people don't seem to like this. Boo.)
2. Stirling numbers. Here, Knuth and co-authors did away with the ever-confusing "Stirling numbers of the first kind" and notations like "S(n, k)", and promoted the names "Stirling subset numbers" and "Stirling cycle numbers" (with the interpretation in terms of set partitions and permutation cycles respectively), and the notation that is slowly catching on (thankfully). (See the paper or, again, read Concrete Mathematics for more fun.)
2a. There's also another sidenote about rising and falling factorial powers; again this notation is also slowly catching on.
There's also a video on YouTube of him giving a talk on notation from October 2003, where covers much of this (and more): https://www.youtube.com/watch?v=KjbuyB4dQa0 (Someone's notes about this video: https://crypto.stanford.edu/pbc/notes/misc/notation.html)