I have read a lot about the SB-800/SB-900 but I am on a low budget (around ~120$). Is there an external flash that I can have to start learning to use it? I do not need TTL compatibility so it can be an older model. I have a Nikon D5000.
I also have the Nikon D5000 and have been very satisfied with YongNuo speedlight. I used YN-460II and recently bought a YN-560.
Both are very good, with easy manual adjustment controls. The YN-560 is bulkier but have electronic zoom feature so this will let you do more experiment with your lighting setup.
I found that I'm not missing the TTL feature as I can guess and required flash power and quickly fine adjust with the button control.
YN-460II costs $50 while YN-560 is $85, so you can get them cheap.
One of the best manual flash options is the LumoPro LP160, but it's a bit outside your price range at $199.
One of the other options that is well within you budget is the Vivitar 285HV, which is a little older design, but a very good product.
You won't get any of the more advanced automatic options, but these should both work great as a simple external flash.
I use the SB400. It's a basic external flash that, despite being plain, is much more effect than the camera's onboard flash. The front of the SB400 can be tilted upwards at varying increments, enabling you to direct your flash elsewhere, or bounce it off a ceiling. The unit uses two AA batteries, and is compact enough to slip in a pocket until needed.
There are higher end units from Nikon, including the SB600 and SB900. They are bulkier, offer more options (e.g. horizontal pivot) and are progressively more expensive.
For a beginner, the SB400 will be more than enough.
I like the sb-400 because of it's small size. Usually I can just put it in my pocket and attach it when required.
Let me add a newcomer here. Nissin 700 - with the remote commander it is a little over the price range, but the commander can handle 21 flashes in 3 groups. It is a decent relatively cheap system.
I use Sony and my choices are very limited - got so far 2 of the units with another 2 being delivered after Christmas. I am very happy with them.
If this is your first/only flash, I wouldn't be so quick to dismiss TTL as a necessity, especially since in the last few years, a number of 3rd-party flashes offer iTTL compatibility within your price range. While a lot of folks read the Strobist and assume they'll never need TTL, having at least one flash that is capable of being used for on-camera bouncing and run'n'gun event/social shooting can be a good thing. Speedlights, after all, are about portability and convenience. There may be times you need the extra control over lighting, but don't want to lug the lighting bag o' gear and mess about with stands and umbrellas.
Consider going with a used Nikon flash. A used SB-600 can be found within your budget. It lacks a few features of the SB-700 (commander capability, 360 swivel, SU-4 mode, etc.), but it's completely iTTL compatible, an has M mode, so can be used off-camera with manual triggers (see: What features should one look for when selecting a flash?).
In my opinion, manual-only cheapies are best for 2nd, 3rd, and 4th, etc. lights that you're going to use exclusively for off-camera studio-type lighting. And even then, you probably want to spend a little more to get a model that offers power control from the camera (typically with a same-brand trigger).
However, with flashes, you can get two of the following three things: rock solid reliability, features, or a low price. So, the units that fall into your price range are likely to have a lower reputation on copy and component consistency and a lower resale, so be careful who you buy from and whether or not you're mailing a unit back to China for warranty repair. But a number of 3rd party brands make TTL-capable flashes. Metz, Godox, Nissin, Sigma, and Phottix might be a little too expensive, but Neewer, Yonguo, and Shanny might come within range. And if you really truly want to go manual-only, then Lumopro is a little higher than your price point, but rock solid, and Cactus can be added to the list.
Listing a specific model is a relatively exercise in futility these days, since so many new models are constantly appearing--especially at the low-budget end of the spectrum. Flash technology is rapidly evolving and has been since 2006 when the Strobist blog first began. Trying to shortcut your research is liable to get you recommendations for older models (as the other answers here show) that made sense at the time, but have been wildly superseded in terms of features you can get.
For example, the manual-only YN-560 went for around US$80 when it first came out, but today there's a YN-560IV that goes for roughly the same amount, with a myriad more features, a built-in radio receiver and much better UI. There's also a YN-568EX for ~US$100 that does iTTL, and can be used as a CLS slave (see also: What are the Yongnuo flash naming conventions?).
Flash Havoc is a great resource for researching a particular model and finding out what new models have been released.