How to display date with human language like "Today at xx:xx pm", "Yesterday at xx:xx am"?

The reason it's blank is that your date format only has time components. Combined with .doesRelativeDateFormatting that gives you the empty string. If you want that custom time format, I think you need separate formatters for the date and the time:

let now = NSDate()

let dateFormatter = NSDateFormatter()
dateFormatter.dateStyle = .MediumStyle
dateFormatter.doesRelativeDateFormatting = true

let timeFormatter = NSDateFormatter()
timeFormatter.dateFormat = "h:mm a"

let time = "\(dateFormatter.stringFromDate(now)), \(timeFormatter.stringFromDate(now))"
println(time)      // prints "Today, 5:10 PM"

With Swift 5.1, Apple Developer API Reference states about DateFormatter's dateFormat property:

You should only set this property when working with fixed format representations, as discussed in Working With Fixed Format Date Representations. For user-visible representations, you should use the dateStyle and timeStyle properties, or the setLocalizedDateFormatFromTemplate(_:) method if your desired format cannot be achieved using the predefined styles; both of these properties and this method provide a localized date representation appropriate for display to the user.

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The following Playground sample code shows how to display your dates in the desired format using dateStyle, timeStyle and doesRelativeDateFormatting properties:

import Foundation

let now = Date() // 2019-08-09 12:25:12 +0000
let yesterday = Calendar.current.date(byAdding: .day, value: -1, to: now)! // 2019-08-08 12:25:12 +0000
let aWeekAgo = Calendar.current.date(byAdding: .weekOfMonth, value: -1, to: now)! // 2019-08-02 12:25:12 +0000

let dateFormatter = DateFormatter()
dateFormatter.dateStyle = .long
dateFormatter.timeStyle = .short
dateFormatter.doesRelativeDateFormatting = true

let nowString = dateFormatter.string(from: now)
print(nowString) // prints: Today at 2:25 PM

let yesterdayString = dateFormatter.string(from: yesterday)
print(yesterdayString) // prints: Yesterday at 2:25 PM

let aWeekAgoString = dateFormatter.string(from: aWeekAgo)
print(aWeekAgoString) // prints: August 2, 2019 at 2:25 PM

Give this a try

let dateFormatter = NSDateFormatter()
dateFormatter.dateStyle = .shortStyle
dateFormatter.timeStyle = .shortStyle
dateFormatter.doesRelativeDateFormatting = true

let date = NSDate()
let dateString = dateFormatter.stringFromDate(date)