While the exact tax laws relating to income and how it is taxed are different depending on which country you are in, the basic principle of taxation is that any income you have is subject to tax unless there is a specific law exempting that particular source of income.
One common exemption which might apply with regards any income you earn from the web is "hobby income" as there are a number of countries where the income earnt from a hobby is not subject to tax (some countries may use different names for it but here it is called hobby income and so that's what I will call it). The law's opinion of what is and isn't a hobby is not necessarily going to allow you to apply it to your web income though. In most cases the legal definition of hobby income for tax purposes is there to prevent your deducting hobby expenses from your other income and is not there to provide you with a way to avoid paying tax on real income received.
When you are running a business (as is generally the case where you are earning income from the web - even where that business is a hobby), the expenses that you incur in running the business are deductible from the gross income earnt from the business and tax will only be payable on the profit from the business. This will mean that the cost of your domain name and web hosting (as well as any expenses associated with getting someone to create the site for you - which depending on how much it cost may need to be depreciated over several years rather than claimed outright) can be deducted from any income that you earn from the web site. Where the deductions exceed the income so that the business makes a loss then it may qualify as a hobby. In fact if your web business makes a loss it is unlikely that you will be able to claim that it isn't a hobby since a business that is completely web based is unlikely to meet any of the criteria it would need to meet to not be considered to be a hobby.
As a hobby your deductions for the business can only be claimed against income from the same business and so with having made a loss the best that you can achieve for tax purposes is to carry forward those business losses so as to claim them against future income from your web business. Only where your business extends beyond the web in a way that allows you to meet criteria to have your business not defined as a hobby would you be able to consider deducting the web expenses against other income you have received.
The situation if your web business actually makes more money than you have expenses will be different. When you make a profit from something then by definition it is not a hobby and the income will be subject to tax. You will of course still be able to claim a deduction for the expenses and may also be able to deduct carried forward losses from prior years but in general once you start making money from the web you will have to pay tax on that income just the same as you do on any other income you receive.
The other thing to keep in mind is that concealing your web income from your local tax authority is not going to work since whatever it is you are using with the site to earn income can be easily seen by anyone viewing the site. While they may have no way of telling how much you earnt from your site simply by visiting it, the tax department will be able to tell that you do have a source of income from the web and if you don't include it in your tax return will be able to take appropriate action with respect to income omitted from your return. If you then don't supply the necessary documentation to show what your earnings and expenses from the web are then they will probably calculate what they would expect you to have earnt based on the declared earnings of others with web income. Most likely that will result in your being taxed on a higher income than you actually earnt from the web plus having to pay fines for having left out that income in the first place. So even if you are not going to declare the losses in your tax return so as to be able to deduct them against future web profits (which if it is a genuine hobby will never arise), you should still keep records of all of your web related income and expenses so as to be able to prove that it is a genuine hobby that never makes a profit if you are ever asked why it wasn't included in your return.
There is nothing special about any income that you earn from the web that allows you to disregard it when filling out your tax return (except where you can prove that the associated expenses were more than the income).
Disclaimer: Note that this article discusses the taxation of web income in general terms. I recommend you see an accountant who specialises in taxation in order to find out exactly what applies to your circumstances and what records you will be expected to keep to justify what is/isn't included in your tax return.