I'm mostly writing this post as a reference point for my guild, but I hope that it might be useful to others as well.
What is Conquest?
Conquest is a rotating weekly event that offers a selection of useful and valuable rewards. In terms of how it works, you can think of it like temporary achievements - complete certain tasks during that week and earn points. The types of activities that award Conquest points are generally very simple and varied, from gaining a level to completing a heroic mission, plus you also receive a small trickle of Conquest points from simply killing mobs or completing any quests at all.
Conquest matters on three different levels:
Personal / Solo
If your character is level ten or above, earning 100k Conquest points in any given week (starting on Tuesday with the weekly reset) will give you a reward. Even if you've never cared about Conquest, even if you didn't know what Conquest even was, you've likely taken part in it by just playing. Earning 100k points on a single character is not hard at all if you play at least a few hours during the week, and the reward would have simply popped up as another mission completion for you. You may have clicked it away without thinking much more than "huh, free stuff".
Guild (non-competitive)
A guild's Conquest score is based on the individual Conquest scores of all of its members combined. At the start of each week, leadership gets to choose a guild target from one of three options (small, medium or large). If the guild reaches this point target within the week, everyone who contributed (by reaching their own personal Conquest target) gets an additional reward after the next reset. Rewards are better based on the chosen size of your target.
Guild (competitive)
Guild targets are associated with planets, and each planet has a leaderboard showing the highest scores of all the guilds that have invaded that planet/chosen that target during the event. The guild at the top of the planetary leaderboard by the end of the week earns a rare achievement and legacy title. This is why guilds may care about scoring extra points beyond their non-competitive target.
Why is my guild asking me to work on Conquest?
In most situations, a guild's officers will choose a guild target that's appropriate for their guild's "natural" level of activity, to ensure that everyone gets their guild reward at the end of the week with no additional effort required. However, there are situations in which leadership might ask members to put in extra effort when it comes to generating Conquest points:
1. You've joined a dedicated Conquest guild. There are usually no more than three or four of these per server, and they are focused on staying on top of the leaderboard each week. They may ask you to score as many points as possible each week to achieve that goal. These guilds will have relatively strict activity requirements and may quickly remove members that become inactive and are no longer contributing.
2. Your guild is trying to move up. As an example, your guild may have been hitting the medium target for a while, but it's been growing/getting more active, and leadership thinks that large yield and its improved rewards are within your reach (but not yet guaranteed). Officers might then choose to go for the large target the next week and ask people to play a bit more than before to push the guild over the threshold.
3. It's Total Galactic War! Most Conquest events only feature three to five planets, meaning that the same few dedicated Conquest guilds will end up on top of the leaderboard over and over. However, there is a rare Conquest event called Total Galactic War, during which every planet in the galaxy can be invaded, meaning there are a lot of opportunities to win first place somewhere. Medium-sized guilds may rally their members around this event and call for a big Conquest push to make use of the rare opportunity to win first place without being dedicated to Conquest all year round.
OK, so... what do I do?
Your guild has asked you to help with Conquest and you want to, but... what do you do? The simple answer is to just play and be active. Almost everything you can do in game generates some Conquest points. As a rule of thumb, organised small group content and PvP will generate points more quickly than just playing solo, but there are plenty of opportunities to score a lot of points as a solo player as well, such as increasing your reputation with a faction or increasing a companion's influence level once per day.
If you do want to learn more about the nitty-gritty of how Conquest works, what activities there are to do and what rewards you can earn, I recommend checking out these guides by Swtorista and Vulkk.