Mostly, you need to figure out if you're struggling to get traffic to the server (i.e. a network connection issue), or if the service is not responding (i.e. some software issue, web server down, etc.)
Tools such as,
can help you work out if the network connection is okay, but they can be blocked by firewalls even if those firewalls let web traffic through, so you need to know how they look when it's working and when it's not working.
Tools such as,
allow you to do DNS lookups locally to make sure that's all working.
As well as running those tools locally, some websites allow you run them so you can rule out your own network connection to the server, for example (chosen at random, no affiliation with these sites),
There are also websites which offer free 'uptime' monitoring, where they will check your site every few minutes or hours and allow you to collect data.
If name lookups work and
traceroute suggest the network is okay, you need to dig down into the web service. That usually entails looking at the web server logs to see if anything untoward is happening on the specific server in question. Is it stop/starting, is the server running out of memory, etc. That's a huge area, and not suitable to a single answer.
To do a full analysis, as the comments suggest, you would need to post logs from the web server before, during and after the issue, along with the exact error messages you were seeing at the client end. But, you asked about tools, and hopefully I've given you some pointers for those.