Update - this answer is for Chart.js 1.x, if you are looking for a 2.x answer check the comments and other answers.

You extend the line chart and include logic for drawing the line in the draw function.

Preview

HTML

<div>
    <canvas id="LineWithLine" width="600" height="400"></canvas>
</div>

Script

var data = {
    labels: ["JAN", "FEB", "MAR", "APR", "MAY", "JUN", "JUL", "AUG", "SEP", "OCT", "NOV", "DEC"],
    datasets: [{
        data: [12, 3, 2, 1, 8, 8, 2, 2, 3, 5, 7, 1]
    }]
};

var ctx = document.getElementById("LineWithLine").getContext("2d");

Chart.types.Line.extend({
    name: "LineWithLine",
    draw: function () {
        Chart.types.Line.prototype.draw.apply(this, arguments);

        var point = this.datasets[0].points[this.options.lineAtIndex]
        var scale = this.scale

        // draw line
        this.chart.ctx.beginPath();
        this.chart.ctx.moveTo(point.x, scale.startPoint + 24);
        this.chart.ctx.strokeStyle = '#ff0000';
        this.chart.ctx.lineTo(point.x, scale.endPoint);
        this.chart.ctx.stroke();

        // write TODAY
        this.chart.ctx.textAlign = 'center';
        this.chart.ctx.fillText("TODAY", point.x, scale.startPoint + 12);
    }
});

new Chart(ctx).LineWithLine(data, {
    datasetFill : false,
    lineAtIndex: 2
});

The option property lineAtIndex controls which point to draw the line at.

Fiddle - http://jsfiddle.net/dbyze2ga/14/

Sharing my solution for chartjs.org version 2.5. I wanted to use a plugin, to make the implementation reusable.

const verticalLinePlugin = {
  getLinePosition: function (chart, pointIndex) {
      const meta = chart.getDatasetMeta(0); // first dataset is used to discover X coordinate of a point
      const data = meta.data;
      return data[pointIndex]._model.x;
  },
  renderVerticalLine: function (chartInstance, pointIndex) {
      const lineLeftOffset = this.getLinePosition(chartInstance, pointIndex);
      const scale = chartInstance.scales['y-axis-0'];
      const context = chartInstance.chart.ctx;

      // render vertical line
      context.beginPath();
      context.strokeStyle = '#ff0000';
      context.moveTo(lineLeftOffset, scale.top);
      context.lineTo(lineLeftOffset, scale.bottom);
      context.stroke();

      // write label
      context.fillStyle = "#ff0000";
      context.textAlign = 'center';
      context.fillText('MY TEXT', lineLeftOffset, (scale.bottom - scale.top) / 2 + scale.top);
  },

  afterDatasetsDraw: function (chart, easing) {
      if (chart.config.lineAtIndex) {
          chart.config.lineAtIndex.forEach(pointIndex => this.renderVerticalLine(chart, pointIndex));
      }
  }
  };

  Chart.plugins.register(verticalLinePlugin);

Usage is simple then:

 new Chart(ctx, {
     type: 'line',
     data: data,
     label: 'Progress',
     options: options,
     lineAtIndex: [2,4,8],
 })

The code above inserts red vertical lines at positions 2,4 and 8, running through points of first dataset at those positions.

I'd highly recommend to use the Chartjs-Plugin-Annotation.

An example can be found at CodePen

var chartData = {
  labels: ["JAN", "FEB", "MAR", "APR", "MAY", "JUN", "JUL", "AUG", "SEP", "OCT", "NOV", "DEC"],
  datasets: [
    {
      data: [12, 3, 2, 1, 8, 8, 2, 2, 3, 5, 7, 1]
    }
  ]
};

window.onload = function() {
  var ctx = document.getElementById("canvas").getContext("2d");
  new Chart(ctx, {
    type: "line",
    data: chartData,
    options: {
      annotation: {
        annotations: [
          {
            type: "line",
            mode: "vertical",
            scaleID: "x-axis-0",
            value: "MAR",
            borderColor: "red",
            label: {
              content: "TODAY",
              enabled: true,
              position: "top"
            }
          }
        ]
      }
    }
  });
};

Have a look here for more Details: https://stackoverflow.com/a/36431041