like the ones in Der Reise http://www.roblox.com/Der-Riese-v0-2-New-Sounds-place?id=17888427
The first thing you want to do is figure out when to award badges. You can do this by integrating the badge awarding process directly in the loop controlling the progression of rounds, and checking whether the round matches the required round to award the badge for.
You can use the AwardBadge method to actually award Badges to players. Here's an example of the AwardBadge method being used, assuming the variable player references the player that will earn the badge and the variable badgeId is the number ID of the badge asset.
To award all players, you can just use a simple loop iterating through all of the players in the game. The line of code above will turn into...
for _, player in pairs(game.Players:GetPlayers()) do game:GetService("BadgeService"):AwardBadge(player.userId, badgeId) end
You can read more about BadgeService on the Wiki.
You'd need the AwardBadge Method. For leaderstats, just add this: (I'm assuming you have the leaderstat script already)
lstat = "Leaderstat name here" badge = 1337 --ID here waveneeded = 5 --wave needed to get badge game.Players.PlayerAdded:connect(function(plr) repeat wait() until plr.leaderstats repeat wait() until plr.leaderstats[lstat] repeat wait() until plr.leaderstats[lstat].Value == waveneeded if not game:GetService("BadgeService"):UserHasBadge(plr.userId, badge) then game:GetService("BadgeService"):AwardBadge(plr.userId, badge) end end)