Note: The techniques described here work best when playing with an organized team or allies that you trust. However, assisting your allies and going out of your way to benefit the team is appreciated at all levels of play! Do a good turn for your ally in a public game, save him from certain death, he might return the favor when it counts most.
The "Greater Good" Role
A well-functioning team needs people in many roles, like those who gank actively around the map and those who farm items for lategame. One of the most key roles to the team's success can be a good supporting player, one who opts to make their plays for the overall benefit to their team, rather than in an effort to get the biggest items possible for themself. While it might sound boring to let someone else get the glory, there's actually a lot involved in being a quality support player.
Who Plays Support?
The best support heroes are generally casters that do not need many items in order to fulfill their role in combat. For example, the Warlock is a classic support hero, since he has a large amount of health and mana by default and all of his skills are incredibly effective on their own, he does not require large items to back them up. Other casters like Lich, Lina, and Crystal Maiden also often fall into this role. It really depends what heroes are on your team, the one that needs items the least should generally take the biggest part in the supporting role.
Any good team needs a chicken from the start! Don't hesistate to buy one and share it with them, they will appreciate it immensely. It takes a lot of time to walk back to the base to buy items, having a Chicken available means that your entire team can stay in their lanes longer and keep the pressure up on your opponents. The Flying Courier upgrade is incredibly useful as well, as it dramatically reduces the transport time. If your chicken is seeing a lot of use, grab the upgrade, everyone will thank you for it!
While map control via Warding is a whole topic in itself, remember that Wards cost money, so somebody has to step up and buy them. If you've got some gold to spare, grabbing the team a pack of Wards is a great investment. If it saves just one ally from one gank, the pack has already paid for itself. Also, Wards often have to be placed in dangerous territory, so the hero that goes to set them is at a good bit of risk. Chances are, the hero sent to place them will be the heavy support, as they are the most expendible, but it can't hurt to convince an ally or two to come with you for safety.
The most enjoyable part of being a support hero is that you'll get to be in nearly every fight for the entire game. That's a lot of action and certainly more than most of the other players on your team will get to see. Keep an eye on your minimap and a Town Portal Scroll on your hero at all times, if you see that your allies are getting into a fight, immediately teleport to join them. Nothing turns a 2v2 battle around like a third hero teleporting in to help. If you show up to lots of fights, your team will win lots of fights, which leads to good things all-around. This is one of the most important aspects of being a support hero -- while you can ask your allies to chip in and help you buy Couriers or Wards, but there is no substitute for you being an active presence in combat. Active and alert support heroes win games, learn it, love it, play by it.
Some opponents can be cautious and hard to gank, but it's often hard for people to resist the urge of taking what looks like an easy kill. Acting foolish and wandering into enemy territory "alone" can be an excellent lure. Ask a couple of allies to lurk in the woods near you, when the enemies come to take you out, throw everything you've got at them and then watch your allies clean up. It's a bit dangerous for you, but with some practice, it's not too difficult to survive. Remember, even if you die, if your allies pick up a couple of kills in exchange, trades like that will win your team the game.
Sometimes things just go badly. It's going to happen, nothing that can be done about it. If you're with an ally and both of you get ganked, try to get a realistic grip on the situation. Does it look like both of you can survive? If not, figure out which of you needs to survive the most. If that's your ally, do everything you can to save them. Jump back into the gankers, disable one of them, block another one, then perhaps try to make an escape in a completely different direction. It's not too difficult to be enough of a distraction to let your ally escape. So the next time you get into a tight spot, don't blindly keep thinking, "I've got to escape no matter what," because a suicidal move on your part might save an couple of your allies and be a huge benefit to your team.