Ajira, a Khajiit, is an Apprentice Mage on the bottom floor of the Balmora Mages Guild. In addition to being your contact for initial quests in this Guildhall, she can also train you in Enchant, Conjuration and Alchemy (48% in each), sell potions, and buy potions and ingredients from you (she has 800 gold). If you complete all of her quests, she will be your bestest friend in the whole world (well, she'll like you a lot; whether you return the sentiment is another question). So save your bartering and training until after you have completed her quests in order to save/make some money off of her (believe me, she's doing the same to you with her quests).
On the whole, Ajira is a decent enough friend, but if the way she deals with the people she doesn't like is any indication, you do not want to be on her bad side.
Ajira wants you to gather some alchemy components that she is supposed to be studying. If you followed my suggestions back in Seyda Neen, you already have the four samples (Luminous Russula, Violent Coprinus, Hypha Facia and Bungler's Bane). If you did not, you'll have to go get them. Ajira will tell you to go look in the Bitter Coast region, but all four are readily available in Seyda Neen, so just take the silt strider back and pick them up. Caution: I've had a little trouble gathering Hypha Facia -- the one in Seyda Neen was empty -- but a quick trek outside of town turned up a sample pretty quickly.
Report back to Ajira to receive a few cheapie potions (4 low-quality Restore Health), a boost to her disposition, a boost to your rep with the Mages Guild, and a completed duty for advancement (don't get pushy -- it's not enough for a promotion yet).
Ajira has made a bet with the other Apprentice at the Balmora Guildhall, Galbedir, that Ajira will make Journeyman before her. Galbedir is upstairs and will sell you soul gems and provide a little training. If you ask around the Guildhall, you'll find that everyone knows about the bet and is keeping their fingers crossed that nothing serious will come of it. What Ajira wants you to do is to put a fake soul gem inside Galbedir's desk. It appears to be a simple enough task. Ajira has even told you when Galbedir will be away from her desk so that you can slip upstairs and do the dirty deed.
There is a catch to this, though. You must move quickly to do this without getting caught. As soon as you close the conversation box with Ajira, Galbedir will start heading downstairs to the main room. She will stay in the main room for a few seconds and then go back upstairs. So make like Speedy Gonzalez and get yourself upstairs before Galbedir gets back.
You must put the soul gem inside the desk -- not on the desk.
Once that is accomplished, report back to Ajira for another boost to her disposition, another boost to your rep with the Mages Guild and another completed duty for advancement (this one will probably make it enough to get you promoted).
Now Ajira wants you to go gather some flowers for her studies. Four varieties, to be exact: Gold Kanet, Stoneflower, Heather and Willow Anther. She directs you to the shores of Lake Amaya, but that's a bit more than you're probably willing to take on as a low-level character. Rather than subject yourself to the depredations of the wilderness encounters, make life easy on yourself. You can find all four flowers on the road from Balmora to Pelagiad. If you are a low level character, the road is virtually empty. You can find Stoneflower and Gold Kanet right next to the signpost where the road turns off to Pelagiad's main street. The Willow Anther and Heather can be found growing along the side of the road, though you might have to head toward Seyda Neen to find the Willow Anther.
If you're not feeling that adventurous, you can buy two of them from Ajira (Heather and Willow Anther) and the Clothier (next door to the Alchemist) has three of the four growing in a flowerpot (Stoneflower, Gold Kanet and Willow Anther, but you'll have to steal them and she's right behind you). The whole point being that there are alternatives to hiking out to Lake Amaya to get these.
When you have the flowers, report back to Ajira for a few cheapie potions (6 low-quality Restore Magicka), another boost to Ajira's disposition, another boost to your rep with the Mages Guild and another completed duty for advancement (not enough to get another promotion).
Your next duty is extremely simple, but has a purpose (getting you out of the building). Ajira wants a Ceramic Bowl and gives you 10 gold pieces to go buy one (you can keep the change).
Pop next door to Ra'virr's shop and buy one (he has three in stock, but you only need one). You have to leave the Mages Guildhall in order to activate the next quest, so save yourself the headache and confusion by just following Ajira's instructions rather than picking up the bowl on your own and having it ready for her when she wants it.
Report back to Ajira to find out that someone (I wonder who?) has stolen her reports. Don't let that deter you. Give her the bowl for a boost to her disposition, a boost to your rep with the Mages Guild and another completed duty for advancement purposes (this one is probably enough for another promotion).
If you have left the building after delivering the flowers, the next quest can activate when you talk to Ajira. If you have not left the building after delivering the flowers, she'll just tell you that she needs a bowl and then tell you that she has enough bowls, thank you very much.
Ajira is not the only apprentice who is not above a little chicanery on the way to Journeyman. Galbedir has stolen Ajira's reports on the mushrooms and flowers you gathered for her. If you can get Galbedir disposition up to 70 or more, she'll even admit it and tell you where she hid them. Rather than go through that hassle, just go look on the floor next to the dresser in the sleeping area for one and on the floor between the two baskets on the landing at the foot of the stairs for the other. If you read them you'll get a few tidbits to help with your potion-making (sorry, no boost to your Alchemy skill) and find out that Ajira is not above taking credit for your hard work (or not-so-hard work if you took my advice).
Return the reports to Ajira for a few good potions (1 exclusive Fire Shield, 1 exclusive Frost Shield, 1 exclusive Lightning Shield, and 2 exclusive Spell Absorptions), another boost to her disposition (which should be up near 100 by now), another boost to your rep with the Mages Guild and another completed duty on your way to the stratospheric levels of the Guild.
Ajira will not have any other duties for you until you have become a Warlock. Once you have reached or surpassed that rank, go talk to her again for information that leads to a coupld of little goodies.
The Staff of Magnus is a non-aligned artifact. Many was the Daggerfall player who quested in vain for it (the quest object was supposed to be random, but wasn't, so the staff never showed up). As a weapon it's pretty cheesy (5-15/1-10/5-15), but it's got a decent Spell Absorption (25%-50% for 60 seconds) and an OK Restore Health (1 point per second for 60 seconds) on it, both "Cast When Used."
Ajira doesn't actually have the Staff, but she knows where it is and points you right at it. It's in Assu, a dungeon on Mount Kand. The entrance is northeast of the entrance to the Mount Kand Cavern. The entrance to Assu is a little difficult to see. With your back to the Mount Kand entrance, head down the road to the left until you get to a clump of Trama plants and a fallen log on your right. There is a trail that turns off to the right at that point and the entrance to Assu is at the end of that trail. It shouldn't take more than a minute or so (real time) from Mount Kand to the turn off.
What Ajira doesn't bother telling you is that the Staff's current owner (Dreveni Hlaren, a 20th level Sorcerer) isn't terribly eager to give it up. But of course you're at least a Warlock, so you should have figured that out on your own by now.
The cave itself is no great shakes, but you'll have to contend with 3 Daedroths (individually, if you're careful) and an 18th level Sorcerer named Fistelle before you can get to Dreveni Hlaren and her pet Frost Atronach. A levitate potion or spell would be extremely handy for getting to Hlaren. While the Frost Atronach is keeping you busy, Hlaren will most likely summon a Bonewalker before she has to fall back on weapons. Report back to Ajira after completing this quest for a 1-point Reputation boost.
After completing Ajira's quest for the Staff of Magnus, Ajira's next little secret is the whereabouts of Warlock's Ring (Fortifies Speed 10-20 points for 30 seconds and Reflects spells 10-20 points for 30 seconds). As with the Staff of Magnus, Ajira tells you where it is and that's about it. This one is a bit more complicated, though.
Warlock's Ring is in the possession of Vindamea Drethan (another 20th level Sorcerer) in Ashirbadon, a cave on a small island west of Bal Fell. Like the staff, the current possessor is not overly eager to part with it. Unlike the Staff, though, Ajira's directions leave a lot to be desired. Perhaps she has been dipping into her own potions a bit too often? Anyway, contrary to what your Journal and Ajira will tell you, Ashirbadon is not west of Bal Fell. It's EAST of Bal Fell.
You'll hit a Frost Atronach or two in the entrance tunnel and then you'll come to a BIG drop-off where you can see the Dremora on the other side. If your Marksman skills are up to it, I'd recommend nailing this guy with missile fire before dropping/slowfalling/levitating down the water. The entrance to Vindamea's area is actually just a little to your right (hiding behind a stalactite), so if you can levitate, you can bypass the Dremora (there are a few potions and such in the chest behind him). You'll find a Storm Atronach just before you get to Vindamea.
The cave is relatively straightforward, but has a lot of verticals to deal with. If you have a high Acrobatics skill, you should be able to just jump up to where you need to go. Otherwise, you'll find a levitate potion in the chest behind the Dremora, which will help you get in. And you'll find another levitate potion on a table behind Vindamea, which will help you get out again. Report back to Ajira afterwards for another 1-point Reputation boost.