Trial Mode is a separate game mode that allows you to challenge your characters by going through 100 stages of baddies, culminating in probably the toughest battle in the game at the end. You can access Trial Mode from the main menu, where you will be given the option to load one of your main game's saves, where you can try out the stages with your current stats and even guests, if you happen to have them in the main game at that point.
While you fight the same enemies as the main game, there are a few differences between them. Some of them have different stats, but they will always retain the same moves they did from their main counterparts. However, all of the enemies don't drop any loot nor give any experience, but they will usually have new items to steal and you will get License Points from them. In fact, there are some pieces of equipment that you can steal/obtain from chests in Trial Mode a lot easier/earlier than in the main game.
Whenever you begin Trial Mode, you will have to load up a main game save and you can get started. The game autosaves at the beginning of each stage and any items stolen/acquired will need you to finish that current stage before you can bring them over to your main game file. Once you have grabbed the items you want, make sure you finish your current stage to trigger an autosave, then you can either die to the next stage or simply pause your game and exit to the title screen. From there, go to Load Game and load the Trial Mode save, which will bring all of the items you earned back into the main game.
Note that doing this and saving over that slot (it's the autosave one) will reset your Trial Mode progress. For every ten stages you complete, you will receive a reward and will be able to make a hard save. This will allow you to continue from that threshold at any time, as long as you don't accidentally overwrite it. Of course, you can't leave and restock your items or gain some levels, so you are essentially stuck if your goal is to clear Stage 100. However, if you want multiple items from a foe at, let's say, Stage 25, then you will need to start over from the beginning of Trial Mode. Of course, with knowledge and x4 speed, it's possible to blaze through the lower stages of Trial Mode with relative ease.
Preparing for Trial Mode
Trial Mode is no joke, especially once you get to Stage 80 and after, since you will be fighting some of the more challenging battles in the game at that point. Since you basically need to finish this mode in one go, it's a good idea to make a lot of preparations, especially since you can't leave to restock or grind your characters, unless you want to start over from the beginning. It's important that your characters can withstand a lot of punishment, so you will want to level them up as much as possible. As an example, one of the strategies for the final stage makes it easier at the max level of 99.
Also, having a max level will help in limited the number of items you use during the mode. This is why attempting to do Trial Mode after finishing the main story is ideal, since you will likely have enough money to purchase 99 of every status-healing item, Phoenix Downs and even Potions/Hi-Potions/X-Potions. MP will also be a big issue, so you will need a healthy stock of Ethers and Hi-Ethers, the latter of which can be bought from the hidden merchant in Necrohol of Nabudis; there are a few stages where your mana will be drained, so you will need them to recover.