People have had to compute things -- process numbers -- for a long time.
It's necessary to work with numbers to deal with accounts, from figuring out billings and payroll down to making change in a store; but more generally, it's necessary to work with numbers to organize any large-scale project.
Those of you who are not as comfortable with working with numbers may be pleased to know that for a lot of human history, very few people were able to work with numbers. Technology for working with numbers includes our number representation system, like how you make an eleven like one-one (write 11), and that "1" means ten whereas this "1" means just one. Actually, this "positional notation" was a major advance in working with numbers, and in mathematics generally.
But I'd like to focus specifically on machinery. Here are some things which I think were particularly new, or that demonstrated new ideas, at certain points in the history of computing machinery.
This presentation is organized as a series of web pages because most web browsers get very sluggish when asked to load a number of large pictures. But if you prefer to see it all together, a single-page version of this document is available here.