Generally you will find on any given casino site a page about how to make a deposit with them, listing the kind of methods that they accept as well perhaps as detailing whether they accept players from countries such as the US and the EU. This is a very important section of the site to note, as it could have some serious bearing on whether or not you are able to play in the first place; remember that not all sites accept US players, so if you are from this country then make sure that you check beforehand. The next thing to do is to chose a method of payment, and there are many which you can look into.
The best advice that can be offered to you here is simply to look around at what is on offer, as there are many options and you will find them to differ in certain ways. Generally speaking, what you are looking for is a service that will allow you to transfer money from your bank account into an online wallet, and then from that wallet into a casino site. PayPal is the most famous example of an electronic wallet that exists, but remember that they generally do not allow you to make payments to casinos and will not be accepted by most sites. It is best to find out which you would be able to use first and then choose from amongst them rather than trying to find a site that accepts the method of payment that you have chosen. If you need any help working out how to move money into your online wallet, you will find detailed instructions available on the site that you have chosen which will walk you through the whole process in an easy to follow manner.
Once you have figured out how to make a deposit into this wallet, you then have to return to your casino site and choose to move this money into the account that you hold with them. Normally this will be a simple process, and a calculator will show you how much you will receive in casino credits to help you understand the whole process better. It can then be as simple as a click of a button to move funds into your account, and you will be able to withdraw them in exactly the same way to make things much easier for you.