For example, running wget https://www.dropbox.com results in the following errors:
ERROR: The certificate of `www.dropbox.com' is not trusted.
ERROR: The certificate of `www.dropbox.com' hasn't got a known issuer.
Looking at current hacky solutions in here, I feel I have to describe a proper solution after all.
First, you need to install the cygwin package ca-certificates via Cygwin's setup.exe to get the certificates.
Do NOT use curl or similar hacks to download certificates (as a neighboring answer advices) because that's fundamentally insecure and may compromise the system.
Second, you need to tell wget where your certificates are, since it doesn't pick them up by default in Cygwin environment. If you can do that either with the command-line parameter --ca-directory=/usr/ssl/certs (best for shell scripts) or by adding ca_directory = /usr/ssl/certs to ~/.wgetrc file.
You can also fix that by running ln -sT /usr/ssl /etc/ssl as pointed out in another answer, but that will work only if you have administrative access to the system. Other solutions I described do not require that.