JavaScript onClick addClass

You can simplify your JavaScript to:

Fiddle

function dodajAktywne(elem) {
    // get all 'a' elements
    var a = document.getElementsByTagName('a');
    // loop through all 'a' elements
    for (i = 0; i < a.length; i++) {
        // Remove the class 'active' if it exists
        a[i].classList.remove('active')
    }
    // add 'active' classs to the element that was clicked
    elem.classList.add('active');
}

If you pass the parameter this in your HTML to:

<header class="mainheader">
    <!-- Obrazek tutaj-->
    <nav>
        <ul>
            <li><a id="a-home" onclick="dodajAktywne(this)" href="#">Home</a>
            </li>
            <li><a id="a-omnie" onclick="dodajAktywne(this)" href="#">O mnie</a>
            </li>
            <li><a id="a-kurs" onclick="dodajAktywne(this)" href="#">Kurs</a>
            </li>
            <li><a id="a-kontakt" onclick="dodajAktywne(this)" href="#">Kontakt</a>
            </li>
        </ul>
    </nav>
</header>

Note: I've changed href attribute to #, you will have to change it back to your .html pages

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Alternatively, you can do this without JavaScript, using CSS's :focus:

Fiddle

a:focus {
    color: blue;
    background-color: #cf5c3f;
}

This code will highlight the navigation tab without needing to include onclick in the HTML, though it doesn't remove the background color if another tab is clicked.

    document.onclick = function(event) {
         var target = event.target || event.srcElement;
         target.style.background = '#cf5c3f';
        };

https://codepen.io/mdmcginn/pen/BwrNaj/