The three tiers differ in one thing that matters: how much of a server you get, and therefore who you share it with. Everything else, price, control, performance, and how much administration you take on: follows from that.
Shared hosting
Many accounts on one server, each isolated from the others but drawing on the same processor, memory and disk.
You get a control panel, and the host handles the operating system, security updates and hardware. You cannot install system software or change server-wide settings, because those affect everyone.
It suits brochure sites, blogs, portfolios and small shops. The large majority of websites. The limit is not usually storage but processing: shared plans cap how many PHP processes you may run at once, and a busy site queues behind itself.
VPS
One physical server divided into several virtual ones. Yours has a fixed allocation of CPU and memory that is not shared, and you get root access.
Root access means you can install anything and configure the server as you like. It also means the operating system is now your responsibility: security updates, service configuration, and recovery when something breaks.
A VPS suits a site that outgrew shared hosting, an application needing specific software, or anything where consistent performance matters more than the lowest price. Ahosting's VPS hosting starts at $8.79/month.
Dedicated server
An entire physical machine. All of the resources, all of the time, and no neighbours at all.
You get complete hardware control and the strongest isolation available, which matters for regulatory requirements as much as for performance. You also take on everything a VPS involves, at a larger scale.
It suits genuinely heavy workloads, applications with hardware-specific needs, and situations where single-tenancy is a requirement rather than a preference. Ahosting's dedicated servers start at $96.75/month.
The line that actually decides it
Not price, and not performance. Who maintains the operating system.
On shared hosting that is the host's job, and it is genuine work you never see: security patches, service configuration, recovery when something fails.
On a VPS or dedicated server it becomes yours. Root access is not a feature you receive; it is responsibility you accept. A server nobody patches is a liability regardless of how fast it is.
So the honest question is not "which is more powerful" but "am I willing to maintain a server, or pay someone to". If the answer is no, shared hosting is the right tier even for a site that could technically use more.
When to move up
Three signals, and only the first is common.
The site is fast at night and slow in the afternoon. You are hitting a resource ceiling. The PHP processes your plan allows are all busy and requests queue. Caching helps a great deal here first; if you are consistently at the ceiling with caching working, the site has outgrown the plan.
You need software the plan does not allow. A specific service, a language runtime, an unusual configuration. This is a hard blocker instead of a performance question.
Isolation is a requirement. A regulatory or contractual obligation that your workload not share hardware.
What is not a signal: a slow site with no caching, or a site that is slow because of one expensive plugin. Moving to a bigger server makes those cheaper to ignore rather than fixed. For telling the two apart, see optimizing WordPress performance.
Reseller hosting, the fourth option
If you host sites for other people, clients, or several businesses you run: reseller hosting gives you a block of shared resources you divide into separate accounts under your own brand.
You get WHM to create and control accounts, and each client gets their own cPanel. The host still maintains the server, so you get administrative control over accounts without the operating system work. Introduction to Reseller Hosting picks it up from there.
Starting point
Start on shared hosting unless one of the three signals above already applies. It is inexpensive, it is enough for most sites, and moving up later is a migration instead of a rebuild.
The mistake that costs money is buying a VPS for a site that would have run comfortably on shared hosting, and then not maintaining it.
If the VPS tier is where you are heading, Introduction to VPS Hosting goes into what you get and what you take on.
Once the tier is settled, the provider still matters more than the specification. How to Choose a Hosting Provider sets out what to ask.
Before moving up a tier, it is worth confirming the tier is the constraint, because most slow sites are not. How to Tell Whether a Slow Site Is the Server or the Site walks through the check.
Decide by the constraint, not by the tier
Comparing the three types on their specifications produces a choice made on the largest number rather than on what the site needs. The useful question is which resource runs out first at your current traffic, and whether the answer is something a larger plan changes. A site limited by how it was built performs the same on any of them. Establishing the constraint first turns the comparison into a decision rather than a preference.