Is my site being attacked? Suhosin simulation, very strange activity in IP Log
Any website accessible on the internet gets pounded by automated attack attempts like this all the time. You can expect your server logs to be filled with this sort of thing. Most are nothing to worry about.
The best thing you can do is have someone monitor the security of your site for you. Whomever designed your site should have been following best practices, which take care of most security issues. If you are buying a web hosting plan (vs a VPS or dedicated server), then your web host will generally take care of security from the server standpoint.
Of course, there are always ways to attack a site. Keep an eye out for defaced pages. If you are storing any customer data, you should hire an expert.
It is just bots probing for vulnerabilities. Happens to pretty much everyone.