(I'll resist the temptation of making the pun that you appear uncertain how to proceed)
You will own the copyright in those elements of the web site which you created, so things like the layout, choice of fonts and colours, and the coding which make it all happen (html, scripting etc), referred to collectively as a literary work. But the client will own the copyright in any data or content which they provided, the flesh on the skeleton so to speak. And of course if the client had paid you as agreed, they would have a implied licence to use the website without infringing the copyright in the part that you created. However since they haven't paid you, the contract is in breach and the implied licence is probably rescinded. I say 'probably' because it will all depend on the detail of the actual contract between you and the client, which could be something as simple as an exchange of emails, but hopefully you have some standard terms and conditions attached to your service, which will also form part of the contract.
However let's assume that the contract is in breach. The only realistic lever you have, given that the client is in a different jurisdiction, is to disable the site.
The best way of doing this is to send a DMCA takedown notice
to the hosting company and wait for the client to contact you. You can do this on the basis that your copyright is being infringed by it being made available to the public without your permission. This is contrary to section 18
of the UK copyright Designs and Patents Act 1988, and for the benefit of your client, section 8(6)(b)
of the Irish Copyright Act 1963.
The hosting company should immediately take the site down and inform the client why this has been done, leaving the two of you to sort out the issue of non-payment.
It is possible that the client has made a copy of the site which could, in theory, be transferred to a new host either with or without the original URL, so you should check this is not happening, especially if the client doesn't seem too concerned about the original takedown.
Unfortunately you cannot pursue the client through the Irish small claims court
, either online or in person, because the system does not allow businesses to pursue claims for payments from private individuals, only the other way round. The only option would be to sue through the Irish District Courts which would involve considerable expense. So let's hope that getting the site taken down has the desired effect.