If you're a small organisation there are 3 options for taking donations through your website. This is a note on the technical choices, not marketing psychology.
Option 1 is very simple, just publish your bank account details on your website. If you have fairly committed supporters, you may not need any more than this, just follow Union-news example.
Using a Charity Fundraising Site
Option 2 is to subscribe to a Charity Fundraising site like Just Giving, set up your own donation page, and link it to your website with one or more strategically placed big 'Donate' buttons. The advantage here, is that someone else is looking after the technical and financial side for you. It's worth reading this article: Charity Fundraising Sites which has some useful explanations and cost comparisons. Most are advertised as free to donors, but there may be a subscription charge, a one-off setup fee, and check the transaction charges. Virgin Money Giving for example, charge £150 signup +2% + 1.45% on card and 6% on Paypal handling - check their Fees page for up to date charges. Also check out PayPal services for non-profits - registered charities with a Paypal business account, pay 1.4% +20p per transaction, with no monthly fees.
Most of these services will provide you with the code for a Donate button to add to your website. Or, make your own / ask your web person. If you are running a Wordpress site, there are plugins to help, like PayPal Donations. Where to place the buttons? If you are asking for donations for your organisation, then place a button in the website header, or somewhere that is fairly constantly in view but not in the way. Examine your website on your phone, imagine you're a user, and think about were the best place might be.
Setting up fundraising on your own site
Option 3 is where you want to run targetted fundraising campaigns on your own website. Maybe you want to build your own landing pages for email campaigns, add custom donation forms, and be able to donate directly from your project pages. You can use the PayPal business account option for handling the money transactions. If you are running on Wordpress, check Give - WordPress Donation Plugin (there are others). You may also need to upgrade your website security, and add more functionality like monitoring, analytics, membership and so on. So, it's not trivial, and it would need looking after, which means more day-to-day people resources.