Pushchairs & Prams. A pushchair is a chair on wheels designed to be pushed. It is a generic term that could include buggies, but generally suggests more padding, and so is suitable for older babies, who are able to sit, but not able to walk. A modern pushchair with three wheels, but the number of wheels is not important.
Many modern pushchairs are convertible. They can be configured as a pram for young babies. Then converted into a pushchair when the baby can sit. And they can fold like a buggy. And so there is overlap in meaning, and words may be used imprecisely to mean “the thing I push the baby in”. So, for example, you may hear people using buggy for a folding pram.
Other terms: Stroller and Baby Carriage are not used in British English (except as an Americanism, several shops seem to be using the term stroller now, so it may see greater use in the future). Wheelchair is a chair on wheels designed to be moved by a disabled user.
A pram is a baby’s bed on wheels (the number is irrelevant, but 4 is typical). It is intended to for young babies to lie in. This black pram is in very traditional style.