AES with different modes of operation
Modes of operation are generally supposed to be independent of the underlying block cipher. They generally have a proof of security showing that the security of a system using said mode reduces to the security of the block cipher. However some modes, such as CTR, don't work well with block ciphers of short length (aka, old ciphers) and can leak information. Popular modes of operation should work equally well with all modern block ciphers.

You wouldn't use RSA in a mode of operation. Typically, RSA might be used to encrypt an AES key and AES would then be used in a mode of operation on the data.
AES is more secure than DES. There are attacks that leverage weaknesses DES's inner workings, while AES stands strong. AES also uses a larger block size (128 vs 64 bits), and small block sizes themselves have inherent weaknesses. Also, DES should not be used by itself, if used it should be used as TDES (triple-iterated DES) and TDES is very slow in software. (That's not a security feature, but it is a pragmatic issue.)

The provided link does mention the block size and speed issues. (And I think it was implied that DES was inherently weaker than AES, given that the question also asked about TDES (that is, DES iterated 3 times, which doesn't have the same weaknesses as DES).)


Briefly on RSA vs AES: They fulfill very different roles. RSA isn't "better" than AES, just useful in areas AES isn't (aka, asymmetric encryption). Conversely, RSA is much slower than AES and has many more caveats in implementation than AES, meaning it's easier to screw-up.