Did people just ditch one for the other all of a sudden? Or was there a gradual change? If it was gradual, how is it that we can tell one from the other?
Did people just ditch one for the other all of a sudden?
No one stopped speaking Old English and then began speaking Middle English. Rather, Middle English is just Old English with some changes. Languages exist on a continuum, Old English, Middle English and English are considered three separate languages, but if time were taken away, they are one language, albeit(with time) at different time periods of its life.
The reason you usually see the start of Middle English incorrectly dated at 1066(it actually began in the late 1100's/early 1200's) is because: 1) ease of memory, and 2) misunderstanding of what was actually happening in English. Most people assume English and French just mixed together like ingredients in a soup, but that never happened.
So now, I'll get this out of the way: The Normans did not cause Old English to become Middle English. They caused late Old English and Middle English speakers to adopt vocabulary from them. Languages do not change because of vocabulary alone, they change because of grammar(the backbone of language), which the Normans had little to no influence on in English.
What actually happened was a bit simpler (and probably more boring): one feature of Germanic languages is the loss of the inherited free Proto-Indo-European accent, and the innovation of stress always falling on the first or root syllable in a word. An unfortunate side-effect of this shift in accent had very long-term consequences for the Germanic family- they started losing their word-endings relatively quickly.
If it helps you to picture what was going on: think of Latin and how important all the word endings are to its grammar, and how the accent in ancient times always fell towards the last two or three syllables in the word, usually among the endings. Now imagine that the accent became fixed at the front, and resulted very quickly in half of those endings disappearing, and the other half collapsing into one or two forms.
In Old English, these very changes had both already happened, and continued to happen. You see that Old English went from having a grammar that was more based on its word endings, for instance: -um, -an, -as, -e, -o and so on, to a grammar that had to restructure itself to do without them. These are the formal beginnings of the very strict word order you see in Modern English: the word endings couldn't provide grammatical meaning anymore, they were either gone or too similar, so grammatical meaning was picked up in how we arrange the words in sentences.
Only a few relics of our earlier system of word endings still exist: the plural -(e)s (as in bugs) comes from the strong masculine plural ending -as, the possessive clitic -'s (the king**'s** horse) comes from the masculine genitive ending -es, the third person verbal ending -'s (he takes) is the innovative replacement for earlier -(e)th, and so on.
As for Middle- to Modern English, /u/Not_A_Facehugger is correct, this is generally marked by the great vowel shift, albeit with further(though less shocking) changes to grammar.
Both were for the most part a gradual change but marked with a specific event. The first, old to middle, started to happen when Normans invaded in 1066 Which brought parts of French dialect that were eventually accepted into the language^1. The second change Happend in the 1500 and started with the Great Vowel Shift^2 in London England^3.