How to extract and access data from JSON with PHP?

This is intended to be a general reference question and answer covering many of the never-ending “How do I access data in my JSON?” questions. It is here to handle the broad basics of decoding JSON in PHP and accessing the results.

I have the JSON:

{
    "type": "donut",
    "name": "Cake",
    "toppings": [
        { "id": "5002", "type": "Glazed" },
        { "id": "5006", "type": "Chocolate with Sprinkles" },
        { "id": "5004", "type": "Maple" }
    ]
}

How do I decode this in PHP and access the resulting data?