Add an item to list in host web, from an app part

You can use XsltListViewWebPart which provide more OOTB look and feel while adding to the list. Provide the ListUrl property with the list you want to add items to

<WebPartPages:WebPartZone runat="server" FrameType="TitleBarOnly" ID="full" Title="loc:full">
        <WebPartPages:XsltListViewWebPart runat="server"
            ListUrl="Lists/CustomDocumentLibrary"
            IsIncluded="True"
            JsLink="clientTemplate.js"
            NoDefaultStyle="TRUE"
            PageType="PAGE_NORMALVIEW"
            Default="False"
            ViewContentTypeId="0x">
        </WebPartPages:XsltListViewWebPart>
    </WebPartPages:WebPartZone>

Another easy way to acheive adding new list item to ShareLoint list using Javascript in APP 2013.

Write the following code in your App.js and call "btnSubmit_CreateListItem" event in ADD button click:

function btnSubmit_CreateListItem() {

    //siteUrl will be your host web url.

    //app web
    var clientContext = new SP.ClientContext.get_current();

    //host web
    var parentCtx = new SP.AppContextSite(clientContext, siteUrl);
    var oList = parentCtx.get_web().get_lists().getByTitle('ListName');

    var itemCreateInfo = new SP.ListItemCreationInformation();
    this.oListItem = oList.addItem(itemCreateInfo);
    oListItem.set_item('Title', 'My New Item!');
    oListItem.update();

    clientContext.load(oListItem);
    clientContext.executeQueryAsync(
    Function.createDelegate(this, this.onQuerySucceeded), Function.createDelegate(this, this.onQueryFailed)

    );
}


function onQuerySucceeded(sender, args) {
    alert('Item created: ' + oListItem.get_id());
}

function onQueryFailed(sender, args) {
    alert('Request failed. ' + args.get_message() + '\n' + args.get_stackTrace());
}