SDHC
The term "SDHC," or "SD high-capacity," defines flash memory cards that store anywhere from 4GB to 32GB. You'll encounter four classes of microSDHC cards: class 2, class 4, class 6 and class 8. Class 8 is the fastest, and class 2 is the slowest; class 2 should be sufficient for use with the Nintendo DSi. However, the lower your class, the slower your transfer and loading times will be. SD cards with a lot of files may lag in your DSi.
Using a microSDHC Card
In order to use a microSDHC card with your DSi, you will need to use an SD card adapter. These adapters are in the shape of an SD card, with a small slot for the microSD card. When you insert the microSD card into the adapter, the entire card fits into any SD-compatible slot and behaves like an SD card. You do not need a special adapter for microSDHC cards. The DSi can handle cards up to 32GB, though the DSi uses "blocks" to measure space rather than gigabytes. A block on the DSi represents 128KB; a 32GB card would have 262,144 blocks.
Uses for Expandable Memory
With the Nintendo DSi you can store photos taken with the DSi as well as view them using the DSi photo menu. You can also listen to compatible music files -- though it cannot store recordings made with the DSi -- and store data downloaded from the DSi store. If you have games stored on your SD card, you can also store the game's save data on your card.
Compatible File Types
The Nintendo DSi can only handle photos taken on the system; it does not support any sort of photos loaded onto the card from your computer or from another camera. If you're using the DSi to listen to music, you'll need audio files encoded in the ACC standard, with either M4A, MP4 or 3GP as the file extension.