Will the website work well on phones?
A website that works well on all devices - including tablets and phones - and in all circumstances - eg. outdoors - is called a responsive website.
Almost every website that is built these days is designed and built responsively, but not always very well. A question you should definitely ask your web designer is "When you design the site, will you include a version in the design for laptops / desktop computers and phones? And how about other devices like tablets?"
And before your website goes like, make sure that you and the website developers test the new site thoroughly on popular devices - such as popular phones like the latest Samsung Galaxy and an iPhone - as well as in different browsers - such as Chrome, Firefox, Edge, and Safari.
So when it comes to websites being responsive, it's no longer a question of "if it is", it's a question of "how well it is".