Byte: Today, a byte is almost always 8 bit. However, that wasn’t always the case and there’s no “standard” or something that dictates this. Since 8 bits is a convenient number to work with it became the de facto standard.
Word: The natural size with which a processor is handling data (the register size). The most common word sizes encountered today are 8, 16, 32 and 64 bits, but other sizes are possible. For examples, there were a few 36 bit machines, or even 12 bit machines.
The byte is the smallest addressable unit for a CPU. If you want to set/clear single bits, you first need to fetch the corresponding byte from memory, mess with the bits and then write the byte back to memory.
The word by contrast is biggest chunk of bits with which a processor can do processing (like addition and subtraction) at a time. That definition is a bit fuzzy, as some processor might have different word sizes for different tasks (integer vs. floating point processing for example). The word size is what the majority of operations work with.
There are also a few processors who have a different pointer size: for example, the 8086 is a 16-bit processor which means its registers are 16 bit wide. But its pointers (addresses) are 20 bit wide and were calculated by combining two 16 bit registers in a certain way.