When reading about beer styles, one quickly realizes that there are no simple answers and that there is no consensus on what defines a particular style. Pale ales and India pale ales (IPAs) are no exception, so as is often the case, the response, in order not to be a book, will involve some generalities and oversimplifications.
Here’s the short answer: Pale ale and IPA are alike in that they share British origin, are hoppier than continental European brews, and employ top-fermenting yeast. But IPAs are hoppier and have higher alcohol content than pale ales. Simple enough, but other questions naturally arise: Why pale ale? Why India pale ale? Knowing the answers to these will enrich your understanding of the differences between the two styles.
Pale ale was originally a gloss for any top fermented beer that wasn’t dark. It was so called to distinguish it from most beers made in Britain at the time, which were brewed with mostly amber and brown malts, and were dark brown.
In the late eighteenth century, Britain started exporting pale ale to its colonies in India (it had already been exporting darker beers there since the seventeenth century). As they had done with the exported dark beers, brewers concocted a custom pale ale to withstand the long voyage to India. Because of hops’ natural preservative qualities, increasing the hops content helped the beer withstand the six-month trip to India. Hence the name India pale ale, or IPA, for this stronger, hoppier ales.
So what does all this mean when you’re ordering a beer? The general guidelines are:
Pale ales are more balanced between the sweet malt and bitter hops. But in IPAs, malt takes a back seat, which is why you will often hear IPAs referred to as hop-forward. This is especially true for Imperial IPAs and Double IPAs, which have noticeably higher hops content. If you’re looking for balance or something that is safely hop-forward, you’ll probably want to avoid Imperial and Double IPAs. (But you won’t know for sure if you don’t try.)
IPAs have a higher ABV (alcohol by volume).
If your primary concern is food pairing, then pale ales, being more balanced, are a safe match for a much wider range of food types than IPAs. If you’ll be eating something spicy, the extra hops in the IPA and spice will go very well together.
The further west you go—from Britain to the West Coast of the United States—the more intense each style can get. Americans have a reputation for taking things to the extreme (often rightly so), and this includes IPAs, and especially some West Coast IPAs. So if a beer is marketed as a West Coast IPA, you can generally assume it is probably stronger and hoppier than a regular IPA.

Again, this is the simple version of an answer that is not that simple. You can easily dig much deeper if you so desire, and you will quickly find online discussions debating what style is, if pale ales and IPAs are even different styles at all, and so on.
But in the meantime, you are now at least armed to debate some of the broader points of these styles over a pint or two.