ZXDS is an emulator of Sinclair ZX Spectrum for Nintendo DS, written by Patrik Rak. It is steadily reaching a mature stage of development, and it already has some noteworthy features, like accurate emulation of Spectrum 48k, 128k and Pentagon models (including sound), or support for TAP/TZX/PZX tape files, TRD/SCL disk files and RZX playback files, not to mention the ability to download these files online over WiFi. It is available entirely for free, but any donations are always welcome, see the details further down the page.
Just download the archive using the link below, unpack it and copy the entire ZXDS directory and simply run the ZXDS the same way you run any other application (not just its content, but the directory itself) to the root of whatever device you use to run homebrew on your DS. Make sure that even the empty subdirectories are copied properly (in particular, make sure the /ZXDS/Saves directory is created, otherwise the save slots won't be saved permanently to your card). If you wish so, you may then relocate the ZXDS.nds to elsewhere on the card, like to the root itself, to make it more accessible.
The provided binary ZXDS.nds is primarily intended for Slot-1 (DS slot) based cards. Most of these cards now support automatic DLDI patching, so you likely won't have to bother with that at all. Just give it a try and you will see. If it won't work for you right away (meaning you won't be able to see any files to load in the load requester) you might need to patch it with a specific DLDI driver for your card with a DLDI patcher, both of which you can find here. If you need the ZXDS.ds.gba version which is better suited for Slot-2 (GBA slot) based cards, simply follow the instructions provided here. Note that you will likely need to DLDI patch it first in this case. If it still doesn't work, I am really sorry, but there is not much I can do about it. Please do NOT contact me regarding if it works or doesn't work on your card, at least at this point of time.
You should also copy some tape, disk, snapshot or playback files to your card. Either TAP/TZX/PZX tape files, TRD/SCL disk files, SNA/Z80 snapshots or RZX playback files are supported. You can place them anywhere you want, although the /ZXDS/Data directory is looked up first by default. For your convenience, I have included few Spectrum games and demos in the archive itself, so you can at least give it a go. And you can find pointers to much more software further down this page.
Alternatively, if you have WiFi access, you can just rely on the /ZXDS/Data/Online directory and browse the plethora of files available online. If you wish, read the online.txt included in the ZXDS archive for more info.