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standard AES disclaimer: Given the questions you've asked, you should not implement AES yourself in a real-world system because there are lots of security considerations when implementing ciphers. Think of the S-box as a function from byte $\to$ byte. So, to look up the image of $x$ under the s-box transformation, you simply use $S_\text{box}(x)$, which ...
You should think of Rijndael's S-box as a function that maps bytes to bytes, where a byte (octet) is considered to be a member of a finite field of size $2^8$ (with xor as addition). It's not seen as a 16x16 octet array, really. The substitution is then just done byte-wise: every octet in the 4x4 block is replaced by its function value under the S-box ...