Suppose you had an index of just two stocks: ABC and XYZ. ABC has a market capitalization of $100B spread out among 1B shares, while XYZ has a market capitalization of $40B spread among 100M shares. So you could add those together, and get a total of $140B. If that's too unwieldy a number, you could divide by a constant number, say 1B, and get 140. Then any time either of the market capitalizations increase by a dollar, you index will increase by one billionth of a dollar, regardless of which stock increased.
Other than the dividing by a billion part, this is a pretty simple system. But if you're Dow Jones, then you decide to, instead of adding the capitalizations together, add the stock prices together. Right now, ABC is at $100 a share ($100B/1B), while XYZ is at $400 a share ($40B/100M). So you just take the average of those, which is 250. This allows you to have a superficially simpler index, in that all you have to do is look at stock prices. Any time any stock price increases by a dollar, your index increases by half a point. Why half a point? Because you have two stocks. If you had 10 stocks, you would divide by 10.
Now, I said this is superficially simpler. That's because if one of the stocks goes up by a dollar, the index goes up half a point, regardless of which stock is went up. But if the capitalization on one company goes up by a dollar, the change in index depends on which stock went up. If ABC capitalization were to go up $100B, the index would go up 50. But if XYZ capitalization were to go up $100B, the index would go up 500.
A further issue is how to handle splits. Suppose XYZ decides that $400 is too high of a stock price, and they're going to do a 2:1 split to bring the price down to $200. So if you do an average of the stock prices, it will be 150. The index dropped 100 points, just because one of the companies decided to do a stock split. If you don't want your index dropping every time a stock splits, you need to do something about that. If you were smart, you'd realize this "look at stock price" thing, although initially simpler, is really not a good idea, and switch to market capitalization. If you were insistent on continuing your current index, you could adjust all the current stock prices and find their pre-split equivalents; since XYZ has a current split factor of 2, you could multiply their stock price by 2, add that to ABC, take the average, and then you would have the same index as before the split.
But if you're Dow Jones, you decide that you don't like either of those options. Instead of adjusting individual stocks, you'll adjust the index as a whole. The index went down by a factor of 0.6, so if instead of dividing by 2, you divide by 2*0.6=1.2, you'll get back to the previous value. This 1.2 is known as the Dow Divisor. It's calculated by taking the sum of the new prices, multiplying by the previous divisor, and then dividing the the sum of old prices. The sum of the new prices is $150, the previous divisor was 2, and the sum of the old prices was $250. So you have $150*2/250 = 1.2. Every time a stock splits, this Dow Divisor has to be recalculated.
Using this metehod means that the although the value of the index doesn't change at a stock split, its relationship to stock price does. Before the stock split, if XYZ were to double in value, the index would become (100+800)/2=450. If XYZ were to double in value after the stock split, the index would become (100+400)/1.2 = 416.67. Every time a stock splits, its effect on the DJIA decreases.
There are also adjustments to the DJIA whenever a company buys another one, but the details depend on whether both companies were originally in the DJIA, or just the buyer was, or just the bought company was, and it also depends on how the deal is structured. If a company in the DJIA splits into two companies, that can also require an adjustment.