Webpack-dev-server compiles files but does not refresh or make compiled javascript available to browser

Two things were causing my problems here:

module.exports = {
    entry: './src/index.js',
    output: {

        // For some reason, the `__dirname` was not evaluating and `/public` was
        // trying to write files to a `public` folder at the root of my HD.
        path: __dirname + '/public', 

        // Public path refers to the location from the _browser's_ perspective, so 
        // `/public' would be referring to `mydomain.com/public/` instead of just
        // `mydomain.com`.
        publicPath: '/public',
        filename: 'bundle.js'
    },
    devtool: 'source-map',
    devServer:{

        // `contentBase` specifies what folder to server relative to the 
        // current directory. This technically isn't false since it's an absolute
        // path, but the use of `__dirname` isn't necessary. 
        contentBase: __dirname + '/public'
    },
    module:{
        loaders:[
            { test: /\.vue$/, loader: 'vue'}
        ]
    }
};

Here’s the fixed webpack.config.js:

var path = require('path');

module.exports = {
    entry: [
        './src/PlaceMapper/index.js'
    ],
    output:{
        filename: 'bundle.js',
        path: path.resolve(__dirname, 'public/')
    },
    devtool: 'source-map',
    devServer:{
        contentBase: 'public'
    },
    module:{
        loaders:[
            { test: /\.vue$/, loader: 'vue'}
        ]
    }
};