Mail and internet not working on my MacBook Air despite being connected to Wi-Fi

This means that your Mac is recognizing the Wi-Fi connection from the modem. However, the modem is likely not connected to the internet. The first thing to do is turn off the modem. Then wait for two minutes and turn it on again. This resets it and often solves the problem. I hope it works for you.

Re: Mail and internet not working on my MacBook Air despite being connected to Wi-Fi

My MacBook Air is connected to wifi but mail and internet do not work

Apple support have an article online (link below) that may help with that:

See what to do: If your Mac isn't connecting to the internet over Wi-Fi - Apple Support