Like in any other country, if you do remunerated work in Japan (i.e. you are getting paid for your work) you have to pay taxes. Generally, there are two kinds of taxes in Japan that almost everybody will have to pay. If you are self-employed or some such, circumstances might be different for you so make sure to check with the tax counter at your local city hall. There are of course other taxes as well, like the property tax or the consumption tax (VAT), but for now, we will be only discussing the income and resident tax.
Employees will have to pay two kinds of taxes, income and resident tax. The income tax (shotoku zei, 所得税), like you, can guess from the name, is based on your income and usually deducted directly from your income. Usually, your employer will take care of this payment for you and you do not need to submit anything on your own. If this does not show up as a deduction on your salary statements, make sure to ask your employer what`s going on.
The resident tax (to / juu / shiminzei, 都・住・市民税) is also calculated based on your income, but it is calculated based on your income from the year before. Thus, if you just started working in Japan, you do not have to pay the resident tax for the first year. Many companies will handle payment of the resident tax on behalf of their employees as well, but some companies do not so make sure to check with your employer whether they will handle the payment for you or not. One thing to note is that you pay resident tax at the place where you live, not where your work. For example, if you worked in Tokyo but lived in Yokohama, you would have to pay your resident tax to the district of Yokohama that you are living in.
Paying taxes is mandatory and failure to do so is punishable. You will also be asked to produce a statement from your local city or district hall, proving that you paid all your taxes when you apply for a new visa, so make sure to check whether your taxes are paid properly or not.