Edit in JSFiddle

<div id="container">
    <div class="centerme">
        <img src="http://placekitten.com/200/150">
        <div class="overlay">
            <p class="centerme">This is some text, also centered.</p>
        </div>
    </div>
</div>

img {
    line-height: 0; margin: 0; padding: 0;
}

#container {
    width: 375px;
    height: 350px;
    background-color: #eee;
    border: 1px solid #ddd;
}
.centerme {
    margin: 0 auto;
    display: table;
    position: relative;
    top: 50%;
    -webkit-transform: translateY(-50%);
    -moz-transform: translateY(-50%);
    -o-transform: translateY(-50%);
    -ms-transform: translateY(-50%);
    transform: translateY(-50%);
}

.overlay {
    width: 100%;
    height: 100%;
    position: absolute;
    top: 0;
    left: 0;
}

.overlay p {
    background-color: rgba(0,0,0,0.7);
    font-family: helvetiva, arial, sans-serif;
    color: #FFF;
}