As a rule of thumb, cut the pepper right behind the rounded portions leading to the stem and base.
If you cut through the entire pepper, you also cut through the seeded part in the center. Preserving that part makes it easier to remove later.
You could switch to a paring knife to help you maneuver underneath the core as you cut it free.
If you don’t wish to use the top at the moment, you could always throw it away or save it for another recipe.
Remember to cut up the top and bottom portions you removed if you plan on using them. They look different from the rest of the cut pepper, so find a different use for them if the shape or size matters in your dish. They often work well in a sauce or soup.
Rubber gloves are available online, at most general stores, and at other places that stock kitchen supplies. If you’re unfortunate enough to experience pepper burn, try using a little rubbing alcohol or milk to neutralize the oil. If you get it in your eye, rinse only with water or saline.
Since hot peppers are generally pretty small, use a paring knife for better results. You could also use a sharp chef’s knife for larger ones.
If you don’t wish to cut through the core, open the pepper from the top and scoop out the seeds. That is useful if you want to cut the pepper into rings.
The heat in peppers comes from that white core membrane holding the seeds in place. Once you remove it, the pepper loses most of its heat, so leave it in place if you don’t mind spicy food!
Pick up a pair of gloves online or wherever kitchen supplies are sold Hot peppers include jalapenos and chilies. You don’t need to use gloves to cut sweet peppers like bell peppers.
If you’re cutting up a bell pepper, also slice off the bottom end the same way. Doing this gives you more uniform rings. The parts you cut off aren’t good for making rings. Instead, you could remove the stem and save the remaining flesh for a snack or another dish.
Check the pepper for any seeds or white fragments left inside. Pull these out with your knife, a spoon, or your finger. This part is trickier with smaller peppers like jalapenos. Switch to a small paring knife and gently work your knife around the edge of the flesh to remove the core. You could also leave the core in place if you want to preserve the pepper’s spiciness.