Following our analogy, if a single-unit spike is one person speaking clearly, and multi-unit activity (MUA) is the chaotic chatter of the 50 people closest to the microphone, then the Local Field Potential (LFP) is the slow, synchronized rumble of that entire section of the stadium.
You aren't listening to individual voices at all anymore. Instead, you are feeling the collective "mood" of the crowd as they move and react together.
Here is what LFP is, where it comes from, and how it fits into the broader picture of brain recording.
To understand LFP, we have to distinguish between what a neuron receives and what it sends.
Just like MUA, LFP comes from the exact same microelectrode implanted in the brain. The difference lies entirely in how we filter the raw signal.

Filtering splits the raw signal into LFP (slow) and Spikes (fast). Source: ResearchGate
When the raw electrical signal hits the computer, it goes through a digital fork in the road:
See how different filtering changes the data you extract from a single raw recording:

You might be thinking, "Slow, rolling brain waves? Isn't that just an EEG?"
They are closely related, but the difference is scale and placement.

EEG reads from outside the skull, LFP reads from inside the tissue. Source: VectorMine / Getty Images