UITableViewCell checkmark to be toggled on and off when tapped

Try this:

var checked = [Bool]() // Have an array equal to the number of cells in your table


func tableView(tableView: UITableView, cellForRowAtIndexPath indexPath: NSIndexPath) -> UITableViewCell {
    let cell = self.tableView.dequeueReusableCellWithIdentifier("cell") as! UITableViewCell

    //configure you cell here.
    if !checked[indexPath.row] {
        cell.accessoryType = .None
    } else if checked[indexPath.row] {
        cell.accessoryType = .Checkmark
    }
    return cell
}

func tableView(tableView: UITableView, didSelectRowAtIndexPath indexPath: NSIndexPath) {
    if let cell = tableView.cellForRowAtIndexPath(indexPath) {
        if cell.accessoryType == .Checkmark {
             cell.accessoryType = .None
             checked[indexPath.row] = false
        } else {
             cell.accessoryType = .Checkmark
             checked[indexPath.row] = true
        }
    }    
}

To reset all the checkboxes:

func resetChecks() {
   for i in 0.. < tableView.numberOfSections {
       for j in 0.. < tableView.numberOfRowsInSection(i) {
            if let cell = tableView.cellForRowAtIndexPath(NSIndexPath(forRow: j, inSection: i)) {
               cell.accessoryType = .None
            }
       }
   }
}

A UITableView keeps selected state for single or multiple selections. So IMO there would need to be a very good reason for keeping an entire parallel state somewhere. If you want to just change the cell's appearance based on select state, do it in the cell.

In your UITableViewCell subclass, override setSelected like so:

override func setSelected(_ selected: Bool, animated: Bool) {
    super.setSelected(selected, animated: animated)
    self.accessoryType = selected ? .checkmark : .none
}

No need to use any table view delegate methods.

Note: You have to call super.setSelected otherwise the cell doesn't keep the selected state correctly.

Swift > 3.0

func tableView(_ tableView: UITableView, didDeselectRowAt indexPath: IndexPath) {
    if let cell = tableView.cellForRow(at: indexPath) {
        cell.accessoryType = .none
    }
}
func tableView(_ tableView: UITableView, didSelectRowAt indexPath: IndexPath) {
    if let cell = tableView.cellForRow(at: indexPath) {
        cell.accessoryType = .checkmark

    }
}

I solved by using two Swift functions: the didSelectRowAtIndexPath and the didDeselectRowAtIndexPath.

override func tableView(tableView: UITableView, didDeselectRowAtIndexPath indexPath: NSIndexPath) {
    if let cell = tableView.cellForRowAtIndexPath(indexPath) {
        cell.accessoryType = .None
    }
}
override func tableView(tableView: UITableView, didSelectRowAtIndexPath indexPath: NSIndexPath) {
    if let cell = tableView.cellForRowAtIndexPath(indexPath) {
        cell.accessoryType = .Checkmark

    }
}

To make this work properly, add a line of code to your cellForRowAtIndexPath function to select a row when the table view is drawn on the screen, otherwise the didDeselectRowAtIndexPath will not be called the first time you select another row. Like so:

override func tableView(tableView: UITableView, cellForRowAtIndexPath indexPath: NSIndexPath) -> UITableViewCell {
    let cell = tableView.dequeueReusableCellWithIdentifier("cellData", forIndexPath: indexPath) 
    if (some condition to initially checkmark a row)
        cell.accessoryType = .Checkmark
        tableView.selectRowAtIndexPath(indexPath, animated: false, scrollPosition: UITableViewScrollPosition.Bottom)
    } else {
        cell.accessoryType = .None
    }

    return cell
}

Swift 3.0
Using just one function to keep it simple

func tableView(_ tableView: UITableView, didSelectRowAt indexPath: IndexPath) {
    tableView.deselectRow(at: indexPath, animated: true)

    if let cell = tableView.cellForRow(at: indexPath as IndexPath) {
        if cell.accessoryType == .checkmark {
            cell.accessoryType = .none
        } else {
            cell.accessoryType = .checkmark
        }
    }   
}