First, going to explain jungling for anyone kinda lost coming in to read the guide: Junglers are champions that do not lane at the start of a game, but instead fight all the neutral creeps scattered across the map.
Who is a Jungler?
Typical candidates include (but are not limited to):
Warwick, Nunu, Malphite, Amumu, Evelynn, Cho'Gath. Sometimes Master Yi or Tryndamere may also start out in the jungles. A lot of times you can tell if these champions will be jungling by their summoner spell build in the loading screen. If they have Smite, Rally, or Heal, there is a good chance they may be jungling. Keep an eye out for this, and you'll do just fine in identifying a jungler.
So, your enemy team has a jungler?
If your team does NOT have a jungler, this could be a big problem...unless you shut them down early. The biggest way to counter a jungler is to gank them before they can really get started, preferably at one of their Sigil camps in order to grab that powerup to help you gank in your planned lanes as well.
If you are playing as a Tank, and see your opponent has a jungler or someone likely to jungle, forego your standard opening items. Buy Cloth Armor (300g) and 1 Vision Ward. Place the Vision Ward at the neutral buff you expect the champion to grab first -- with physical champions like Warwick or Malphite, this is usually the Lizard; with magical ones like Evelynn, Amumu, and Nunu expect the Golem. They may switch things up, or they may even enter your jungles and grab one of your buffs. This is okay...you're at least prepared for later and they don't have a clue.
If you're playing as something other than a tank, that's fine as well. Get a few of your teammates to come with you *AS FAST AS POSSIBLE* and wait in the brush at the buff you expect your opponent to go to. It's best to get the people from the nearest lanes, unless everyone has teleport -- the less laning you miss, the better.
When the Jungler arrives, let them do a little bit of fighting of the neutral creep (unless they see you first, obviously), but then pop out and start to gank the player. Don't attack the neutral buff creep -- chances are, its aggro is still on the enemy champion and they hit HARD at this early stage in the game. When you kill the enemy champion, try to mop up the buff and use it to your advantage. If possible, leave the baby creep there as well. This prevents that buff from resetting until the jungler goes to clear that camp.
After the start.
Try to keep tabs on the Jungler as much as possible. If you see them pass through your vision, make a note to your team of the champion's level. If you know they're at a buff camp, try to get another teammate to come with you and either gank the player, or steal the buff. Either way, you're slowing them down.
If you ever kill the champion while he is attempting to jungle, use that gold positively, by either giving yourself items to help push your lane in and give a lot of harassment to the enemy champion (preventing them from taking advantage of their Solo lane), or by buying a Ward to set up another gank at a buff creep).
If the enemy Jungler has hit Level 6, there's usually a good chance they'll be attempting Dragon soon. (With Amumu, he can attempt this at 4 or 5 with good masteries/runes, as long as he has a Golem Sigil). If you have someone willing to buy a Ward, place it at Dragon. Killing the player at Dragon *AND* stealing the Dragon while it's weakened is an EXCELLENT boon to your team. Not only did you stop that player and get the bounty for them, you also just got global experience and global gold for your team.
Late game, Junglers tend to start playing with their team, so at this point, there's really nothing to tell you. If you slowed the Jungler down enough, he or she will be underleveled at this stage in the game, giving your team a massive advantage in team fights. If you also took advantage of the shorthanded lane assignments and pushed the towers, you've got an even bigger advantage. Congratulations, you stopped the Jungler.
As an alternative to buying a Ward, you could potentially invest in the Clairvoyance summoner spell. Due to the changes in it recently, this spell is often considered weak, but when it comes to scouting out a Jungler it can be a big help and a cost-saver. Its current implementation makes it perfect for checking buff creeps, and if used well, can help you keep tabs on that pesky jungler without having to waste time out of a lane or precious gold on a 3-minute Ward.
Additionally, if you're laning and haven't seen the jungler in a while, try to stay near the outside of your lane. This will give you more time to react if the jungler comes out to try to gank you. Most often, Junglers will attempt a gank when they: 1) Have Lizard Buff, 2) Hit Level 6 (most of them have great Ultimates for Ganks), 3) See a weakened player pushed pretty far into a lane. If you're taking a lot of damage, make sure to stay toward the outside and a little back. Don't autoattack the creep, but still try to last hit for that precious gold. If you start getting your health back, then you can try pushing again.
If you're playing a jungler as well
I don't know why exactly I skipped over this in my initial post, but here it is:
If you're playing as a jungler, or at least as someone who can clear mini-camps, one of the best ways to stop or slow the enemy jungler is to venture into their jungles and take their camps.
At the start of the match, get a teammate to come with you and wait in the brush at the jungle camp you expect them to open with. (Do your best not to be seen by the opponent). Let the jungler fight the camp and get weakened, then attempt to steal the buff (or even let them kill it, if you have a disable that can hold them in place) Killing the person and taking their buff, then clearing that side of their jungles will be a huge roadblock for them, as well as a head start for you.
Try to venture into their jungles as often as possible to take their buffs, and try to be sure you're the one killing Dragon. Dragon is the big difference between the effectiveness of junglers, because it not only gives you a ton of experience and gold, but your teammates as well.
Doing these things will guarantee your jungling beats out theirs and give your team the edge.