What convert PNG to ICO actually produces
Not a renamed PNG. An .ico is a container format: a small header followed by one or more complete images, and since Windows Vista those images may be PNGs rather than the old bitmap layout. So a proper .ico holds several resolutions at once, and whatever displays it picks the closest match.
This tool writes three of them, at 16, 32 and 48 pixels. Those are the sizes that actually get requested. A browser tab uses 16 on a standard display and 32 on a scaled or retina one, and Windows Explorer asks for 48 in its list views. A single-size icon forces every one of those consumers to rescale, and rescaling a 16 pixel image up, or a 48 pixel one down, is exactly where icons turn to mush.
Non-square sources get cropped, not squeezed
Icons are square. If your source is not, something has to give.
The choice here is a centre crop rather than a stretch. Distortion is far more noticeable than a trimmed edge at these sizes, and a squashed logo reads as broken in a way a slightly tighter crop does not. If the important part of your artwork is not centred, crop it to a square yourself before converting so you control what stays in frame.
Start with something big
Every size in the file is downscaled from what you provide, so the source resolution sets the ceiling on quality. A 256 pixel square PNG or larger is ideal. Coming in at 32 pixels and asking for a 48 pixel entry means upscaling, and this tool will not invent detail that is not there.
Simple, high-contrast artwork survives the trip to 16 pixels. Fine lines, small text and subtle gradients do not, no matter how good the source is. If your logo has a wordmark, the icon usually wants just the symbol.
This is a favicon generator, with one deliberate limit
If you searched for a favicon generator, this is one, and the .ico it produces is the part that actually needs generating. What it does not do is emit the full set of Apple touch icons, Android manifest icons and Windows tile images that some generators bundle in. Those are ordinary PNGs at fixed sizes, so the resizer on this site makes them in one pass, and none of them needs a container format.
Where the file goes
Rename the download to favicon.ico and put it at your site root, so it resolves
at /favicon.ico. Browsers request that path automatically even when your HTML
says nothing about it, and so do a number of crawlers and link-preview services.
Declaring a PNG or SVG icon in your HTML alongside it is still worth doing, since those give you higher resolution where it is supported. The .ico is the floor, not the whole story.
Built in your browser
The PNG rendering and the container assembly both happen locally. No file is uploaded, which for a logo you have not launched yet is a reasonable thing to care about.