It says my browser cannot read HEIC. Now what?
Convert on the phone that took the photo — this page works in Safari on an iPhone or iPad, and on a Mac. That is not a deflection: HEIC decoding belongs to the operating system, Apple's has it and Windows mostly does not, and shipping a megabyte-plus decoder to everybody so that some visitors avoid picking up their phone is the wrong trade.
JPEG or PNG — which should I pick?
JPEG for a photograph: far smaller, and the loss is invisible on a picture of the real world. PNG for a screenshot, a logo, a diagram or anything with transparency: lossless, and actually smaller than JPEG when the image has few colours and hard edges. Automatic looks at the image and decides on exactly those grounds.
Does converting lose quality?
To JPEG, a little, once — it is encoded at high quality and at full resolution, and the loss is not visible at normal viewing size. To PNG, none at all: it is lossless. Neither is scaled down; the picture keeps the pixels it arrived with.
Is my photo uploaded anywhere?
No. The decode and the re-encode both happen in this tab, and the file never leaves the device. The photos people most often need converted are of documents, and a converter that wants a copy of your passport page to change its file extension is asking for far more than the job needs.