It really depends on how many users, and how intensive the application is.
Do you already have the application made? If so, what language is it in?
As far as hosting goes, all 'types' boil down the 3 basic classifications. (All gimmicks aside.)
Shared Hosting - You and other people share space/bandwidth/resources on one physical server/computer. This is cheap and the most common among small-mid sites. This is good for people that don't need a lot of power and just need to serve some basic stuff. Price is usually broken down by how much space and bandwidth you're allotted on the server.
Dedicated Hosting - You and you alone get the full power of an entire physical server/computer. This is much like buying a PC...and you are the only one using it. This is for people who need to full power of the processor and have to look at RAM usage.
VPS Hosting - This is a bit of a mix between the two above. While other people are on the same phsyical server/computer as you. It's not as 'shared' as actual Shared hosting. With a VPS, using software, the hosting provider has basically created smaller dedicated servers on one server/computer. So a lot less people actually share this, you get all the conrol of a dedicated server, plus you'll get dedicated hardware resources like CPU and RAM.
What do you need?
It depends on a couple things like I mentioned above. If the software you are installing is not that resource intensive AND you won't have too many users ('too many' = varies depending on software), you might be able to get away with shared hosting.
If shared hosting won't work for you, it will mostly be because of resource usage. RAM/CPU disk I/O. Then you will have to look for a VPS or dedicated server.
Choosing between VPS and dedicated comes down to how much resources you actually need. Once you figure that out, and can go shopping.
Id also look into cloud hosting if youre thinking of doing stuff like google docs with multiple users. access for everyone would be easily password protected and with cloud you can hide certain things behind a firewall in your cloud so theyre private, so its good for sharing with lots of users but still keeping your own stuff personal
Depends on your users and resource usage. If you get a lot of users, you should get a VPS or dedicated server. If you only get a few users everyday, then get a shared or cloud hosting and then upgrade to VPS or dedicated server as you grow.
The hosting product you choose should correlate to the level of traffic you anticipate, and is irrespective from the size of your application.
For example you can have a mammoth enterprise grade application with 2 simultaneous users during peak times and run fine from a virtual hosting account.
Capacity planning and trend recognition is important, that way your able to upgrade to the next hosting package tier prior to your traffic levels spiking. IMO its usually better to invest your money into well though out marketing rather than to dump it into hosting right away.
Out of curiosity, what sort of budget do you have for this? I think that you would be fine starting off on a shared hosting plan
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If you are planning to run a site just like Google documents and will have more users then dedicated or VPS hosting will be best for you. However it depends the resources used by your sites and users. If you have more users then you will need of dedicated hosting but have less users then go for VPS.