Sounds like someone spread grease on the terminals in a misguided effort to stop corrosion. Even some professionals do that thinking they're doing their customers a favor.
In fact, any type of treatment for corrosion is a waste of money., including the pretty "juicy rings" that look like red and green felt. When the battery is newer, that corrosion is not going to form. As batteries age, the lead flakes off the plates. As that material builds up in the bottom of the case, when it gets high enough, it shorts out a cell, then the battery must be replaced. Before it gets that bad, the lead flaking off leaves you with less and less material on the plates. In effect, the battery becomes smaller electrically, but the charging current going through it stays the same. With that current being pushed through less material, the plates become hotter. That increases the bubbling of gases in the acid. That bubbling is normal to a lesser degree, but when it increases in older batteries, those bubbles tend to float up and pop right under the top of the case. That deposits acid on the underside of the top of the case. From there the acid finds its way out through the gaps around the two posts. That's what causes the corrosion.
What I found while working at a very nice family-owned dealership is if there's no corrosion around the posts, there's no need for any type of treatment. If there is corrosion, that battery is going to fail within about six months or less, and no anti-corrosion treatment is going to stop that. My experience has been those juicy rings, and the grease some people use, is more likely to find its way in between the battery post and the cable clamp attached to it and cause a less-than-perfect connection. Better to just leave them dry like they came from the factory.
To answer your questions, I wouldn't be too excited about cleaning that material off if it is in fact grease. You can connect jumper cables right through that grease, but give your battery a couple of minutes to charge up before you try to start the engine. Trying to start it right away draws the current from the donor car. The clamps on jumper cables can't pass enough current on their own to run a starter motor. After a few minutes with the jumper cables attached, your battery should crank the engine with the help of the donor car.
Here's links to some articles that will explain this better:
https://www.2carpros.com/articles/how-to-jump-start-car-battery-using-jumper-cables
https://www.2carpros.com/articles/everything-goes-dead-when-engine-is-cranked
https://www.2carpros.com/articles/how-to-charge-your-car-battery
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Friday, May 26th, 2023 AT 8:32 PM