jQuery getting previous input value

Given that the input element has an id attribute it should be unique, so you can select it directly without the need to traverse the DOM:

$('#btn-chat').click(function() {
    var inputVal = $('#btn-input').val();
    // do something with the value here...
});

If, for whatever reason, you still want to use DOM traversal you can use closest() to get the nearest common parent to both elements, and then find():

$('#btn-chat').click(function() {
    var inputVal = $(this).closest('.input-group').find('input').val();
    // do something with the value here...
});

If you have multiple elements in your page with the same id attribute then your HTML is invalid and you would need to change it. In that case, use class attributes to identify and select the elements.

$("#btn-chat").click(function(){
    var input_values = $(this).parent().parent().find("input").val();
    alert(input_values);
});

As @RoryMcCrossan has pointed out, if you have multiple elements with the same id attribute, you would need to use a class attribute instead.

$(function() {
    $('.btn-chat').on('click', function() {
        var val = $(this).closest('.input-group').find('input.btn-input').val();
                  //OR $(this).parent().prev().val();
        console.log( val );
    });
});

$(function() {
    $('.btn-chat').on('click', function() {
        var val = $(this).closest('.input-group').find('input.btn-input').val();
        console.log( val );
    });
});

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<div class="panel-footer">
    <div class="input-group">
        <input type="text" class="form-control input-sm chat_input btn-input" placeholder="Wpisz tutaj swoją wiadomość..." />
        <span class="input-group-btn">
            <button class="btn btn-primary btn-sm btn-chat">Wyślij</button>
        </span>
    </div>
</div>


<div class="panel-footer">
    <div class="input-group">
        <input type="text" class="form-control input-sm chat_input btn-input" placeholder="Wpisz tutaj swoją wiadomość..." />
        <span class="input-group-btn">
            <button class="btn btn-primary btn-sm btn-chat">Wyślij</button>
        </span>
    </div>
</div>

<div class="panel-footer">
    <div class="input-group">
        <input type="text" class="form-control input-sm chat_input btn-input" placeholder="Wpisz tutaj swoją wiadomość..." />
        <span class="input-group-btn">
            <button class="btn btn-primary btn-sm btn-chat">Wyślij</button>
        </span>
    </div>
</div>