What are the different ways to find broken links on the website and how to fix them?

There are many ways to find broken links on the website.
As @mohammad Al-Shafie mentioned Google Webmaster tools , SEOmoz .
Some other Software and techniques through which you can find broken links are: - xenulink is desktop software - http://www.linktiger.com using this website - http://www.addme.com/link-checker.htm - http://validator.w3.org/checklink and so on.

Your best friend to find broken links is the Error Log of your web server.
You can use a log analyser [ it depends on which web server you are running ] and it should provide you with a clear report on all pages that were not found.

The first and definitely the most important would be Google Webmasters Tools.
However you could use other tools like SEOMoz.

Well it depends on the size of your website (e.g: how many indexed pages you have in Google)? Now if you have a very big website, try to review all your main/important pages links - even if manually - , online tools may not provide a full review option, but it worth trying, Riyad mentioned many good tools in his answer I suggest you try them all, see what fits your needs.
Now, if your website is few hundreds pages only, tools are perfect, can find you every broken link.
For me with more than 1 Million indexed pages website, I place a tracking code into the 404 custom error page, this way I get the broken link and the source of the visit.
Next step would be to fix those broken links taking into consideration the following points:
1- 301 Redirect each broken links to the appropriate exact content link (individually, dont ever make a global 301 redirect for all 404 request).

2- You will find that some visitors are coming from misspelling direct visits, if the number is high it would be a good idea to create a redirection rule on your server to catch those visits.

3- If visits are coming from external links, it would be a good idea to contact the webmaster and ask for a link modification, this way you eliminate 404 errors while getting a link back in the same time.
Also remember to provide an informative 404 custom error page, where you offer alternative content and a search box for visitors reaching your 404 pages.
Good Luck.

you can check Google Webmaster tools , SEOmoz or the site map

there are many Firefox and Chrome add ons that might help also if you are using WordPress find broken links checker plugin webmaster tools will show you broken links you have in your site