Finding the Temple
Go to the Calm Lands, and seek out a lady riding a chocobo: she is often found by the cliffs to the west, or by the Al Bhed store in the center of the Calm Lands. Approach her and ask if you can ride a chocobo, ignoring the option to train a chocobo for the time being (this is only required to power up Tidus' Celestial weapon). Like Mi'ihen Highroad you will have no encounters and travel a lot faster on a chocobo, however at the Calm Lands it is free to ride.
Riding your chocobo, return to the entrance from Macalania Woods, but instead of going back to Macalania, keep heading slightly further east to find a broken bridge with a chocobo feather lying on the ground next to it. Examine the feather and your chocobo will jump down to a previously inaccessible plateau (to get down examine the next chocobo feather), so continue the short distance to the exit from this plateau to find yourself at the impressive Remiem Temple. Inside, you will find not only a hidden (but currently locked) aeon, but also Belgemine (as long as you fought the Aeon battle against her in the Calm Lands). Around the back you will find an Al Bhed Primer vol. XXIV as well as a chocobo race course, as detailed in the Celestial Weapons page.
Unlocking the Secret Aeon
Your (probably final) secret Aeon is located here: The Magus Sisters. As is the case with secret Aeons there is no Cloister of Trials required to unlock this Aeon, however another challenge does exist. The Chamber of the Fayth is directly accessible at the back of the temple, however it is currently blocked by 3 separate seals.
One of the seals can be broken by the Flower Scepter, which Belgemine will give to you after defeating her Bahamut (you must also defeat the other 4 Aeons before it). For the strategies for these battles see the below section. The second seal is broken by the Blossom Crown, which the Monster Arena owner will give to you as a reward for capturing at least one of every fiend at Mt. Gagazet (both in the cave and outside). The third is broken by obtaining Yojimbo in the Cavern of the Stolen Fayth and obtaining Anima at Baaj Temple.
You may now enter the Chamber of the Fayth to obtain the 3 Aeons (although they are summoned as one) Cindy, Sandy, and Mindy that make up The Magus Sisters. You can see the individual Aeons page to learn more about how these unique Aeons operate.
After you have obtained the airship and visited the Highbridge you will be able to enter the Chamber of the Fayth to find a treasure chest containing a Defense Sphere.
You can only fight an Aeon once you have obtained it yourself, and have defeated all the Aeons before it in the sequence. Battles are fought as normal Aeons battles, in that you only have Yuna to fight with. You will receive the Summoner's Soul, if you do not have it after defeating Valefor, and the Aeon's Soul, if you do not have it after defeating Ifrit (or Valefor if you already had the Summoner's Soul). As mentioned above, you will receive the Flower Scepter after defeating Bahamut. When you defeat all 8 Aeons and then choose to send Belgemine you will receive the Moon Sigil. You will also receive some regular items after each fight.
Valefor has 20,000 HP, and is able to use both Energy Ray and Energy Blast as overdrive. You are probably best off summoning Shiva for this battle, due to her superior speed and ability to heal herself with Blizzara. Simply use physical attacks, use Block if Valefor reaches overdrive, and use Shiva's overdrive when you get it. If Shiva dies, simply summon Bahamut to finish Valefor off using a similar strategy. Any of the hidden Aeons will also easily polish Valefor off. You will receive 4 Lightning Gems for your efforts.
Alternative Strategy: Call one of your summons with an elemental affinity (Ifrit, Shiva, Ixion), attacking on most turns and using their respective elemental magic (Thundara on Ixion, Blizzara on Shiva, etc.) on themselves as healing, since they will absorb these spells. This can be quite useful in these battles, especially if Yuna is under leveled. Subsequent victories yield Power Sphere x4.
Ifrit has 28,000 HP, but otherwise is very similar to Valefor. Just use the same strategy as for Valefor, except using any of the hidden Aeons where available. Upon winning, you will receive 30 X-Potions. Subsequent victories yield Mana Sphere x5.
With 30,000 HP, Ixion is only marginally tougher than the two previous battles, and you will again be able to use exactly the same tactics. Ixion sometimes defends, which is the equivalent of shielding for a turn, or evades which raises his evasion stat greatly for the turn. After your victory, you will receive 10 Chocobo Feathers. Subsequent victories yield Power Sphere x8.
Shiva has 20,000 HP, but much better stats than the other Aeons so far. Any of the hidden Aeons won't have trouble defeating her, and you may find Bahamut's overdrive a good choice. Failing that, using Ifrit will exploit her weakness to fire, and can also heal himself using fire. Ifrit's overdrive will quickly help fell Shiva, upon doing which will get you 60 Mega-Potions. Subsequent victories yield Star Curtain x6.
At 35,000 HP, the fight against Bahamut will be rather different than those against the other Aeons. Although Bahamut sometimes uses a physical attack (an Aeon with a high evade such as Yojimbo will help here), he will mostly charge for Mega Flare. Mega Flare is pretty much guaranteed to kill whatever Aeon you have out, so shield if possible, or ideally defeat him before he gets to this stage. If you don't have Anima yet, Yojimbo would be a good bet due to his high strength and evade. Pay him 1,025 Gil (or more going up in powers of 2 plus one each time if necessary), and you should keep getting Wakizashi hitting for a high amount of damage. If you don't have any of the hidden Aeons, use the general strategy outlined above, although you may need a slightly higher leveled Yuna. You will get the Flower Scepter after this battle, which along with the Blossom Crown will enable you to obtain The Magus Sisters (see above section for details). Subsequent victories yield Mana Sphere x8.
Yojimbo's 32,000 HP is best reduced by coming into the battle with an Aeon already in overdrive (namely Bahamut), or by Grand Summoning that Aeon. Now that you are able to have obtained The Magus Sisters, they can help in dispatching Yojimbo, for doing which you will receive 8 Shadow Gems. Anima can also kill Yojimbo quite quickly. Subsequent victories yield Power Sphere x10.
Anima is as ever a beast of an Aeon, with 54,000 HP and a high strength stat. If you have plenty of money, pay Yojimbo either 2049 or up to 8193 a time to try to ensure he uses his Wakazashi attack. You may run through a lot of Aeons here if you don't have The Magus Sisters to assist you, so if you are still having trouble, come into the battle with their overdrives already filled. You will receive 60 Stamina Springs after Anima goes down. Subsequent victories yield Mana Sphere x10.
The Magus Sisters
In this threesome, Sandy has 35,000 HP, Cindy has 48,000 HP and Mindy has 20,000 HP. As between them they have such high HP, and will get a lot more turns than you, your best bet is to use Oblivion and Mega Flare (come into the battle with the overdrives already filled). Also, it would be in your benefit to kill Cindy first, due to the fact that she is their main healing source. If that doesn't manage to defeat these Aeons, bring Yojimbo out and follow a similar strategy as you used for Anima. Your other Aeons will not be of much use to you unless you have an overpowered Yuna (in which case the hidden ones would have been a better choice, anyway) or if you already have their overdrives filled. Winning this final Aeon battle will net you 40 Shining Gems, and the option to send Belgemine, upon doing which you will receive the Moon Sigil.