How to become landowner

There are two concepts to be aware of for owning land -- purchase and maintenance. Purchasing land remains the same irregardless of whether one is in a mainland or a private island sim, but there are differences between maintenance on the mainland and maintenance on private islands, as well as purchase of entire private islands, which will be covered further down.
===Why must I pay to own land?===
You can picture Second Life's creators, Linden Lab, as "3D content hosting providers". Their business is running a "grid" made up of thousands of computers (providing regions of 256x256 m in size). Think of those as 3D web servers. "Land" is just an abstract representation of disk space, CPU consumption, and bandwidth. The more resources you require, the bigger your "land". That's the same concept as hosting web pages, and paying for more MBytes of disk space, or more GBytes of bandwidth traffic, etc. The more resources you use, the more you pay for it.
Remember, this is Linden Lab's way of making money, since a basic account is for free and you don't pay anything for the SL viewer client (the application you installed) or to connect to Second Life. If you don't wish to pay anything to Linden Lab *ever*, you can rent land from other Residents -- they'll charge you an amount of L$ every week to allow you to set up shops and your home on their land.
You don't have to own land to enjoy Second Life, although Linden Lab hopes to encourage you to become a Premium user and buy some land, so that they may pay their developers to program better features, give us faster servers to run the grid, and so on :)
==== Purchasing Land In General ====
Purchasing land is a one-time cost that occurs at the time you actually claim the land as yours. 
You can purchase land from other residents, by clicking the Find button at the bottom of your screen, then selecting the Land Sales tab. Or you can purchase new land on the Second Life auctions, at http://secondlife.com/auctions
Be sure to compare prices of land that is readily available in-world with the prices on the auctions. Most of the time you'll find the auction prices will come out cheaper, since the readily available land is usually marked up due to its convenience.
To purchase a plot of land that's available, you'll have to travel to where it's located, then right click on the surface of the plot (use 'P' to toggle parcel borders), select About Land. Once you have reviewed the window and are sure you want to buy it, click Buy Land... to make the purchase.
==== Purchasing Mainland Land ====
Prior to making a purchase of mainland, you are required to have some form of payment info set up in your account with Linden Lab. This is to allow Linden Lab to make the necessary arrangements to keep maintenance fees for the land paid. If you have not owned any land on the mainland prior to the purchase, Linden Lab will also alert you to the fact that you need to have a premium account on at least a monthly basis in order to own mainland land. Linden Lab will also prompt you to upgrade your monthly tier in order to cover at least the amount of land you are going to own as a result of the purchase.
==== Purchasing Land In Private Islands ====
All private islands allow their owners to set a covenant: a notecard which adds additional rules on top of the basic Second Life ToS and Community Standards which the buyer must follow if he or she is interested in owning land in a private island.
Covenants may be examined by right-clicking on the soil (actual soil - not prim-based land covers!) of the parcel you are interested in and selecting about 'Land'. 
It is important that you read this covenant carefully prior to making a land purchase or rental in such an island, as virtually all covenants include rules that allow the island owner to repossess land without compensation in the event of a significant breach or continued breaches of the covenant! More often than not, these will be slightly more restricted than the ToS and CS in order to guarantee a certain experience, personalised service and/or quality of life differing from that on the mainland.
==== Purchasing entire private islands ====
The purchase of entire private islands in Second Life requires a premium account or prior arrangements for alternative means of payment with Linden Lab, and can be done at the Land Store (SL account login required).
On logging into the Land Store, a grid map will be displayed that tells you what parts of the grid are ready for a new private island. By clicking on each region in the grid, available spaces may be secured with a reservation. Please visit the Land Store for current pricing.
=== Mainland Maintenance ===
For mainland lands, maintenance is the monthly cost you pay to Linden Lab for the peak amount of land that you owned during that month. This is calculated based on the largest land area you owned for the billable month, even if you divest land that month in sufficient quantities to qualify for lower tier.
A *premium* subscription with SL grants you free maintenance on 512 square meters of mainland land, but does not cover the purchase of the land.
If you want more land than the 512m you get with your premium membership, or if you have a basic membership and want to own some land, you will have to select a "land tier". Land tiers range from 512 sq. meters all the way up to 65,536 sq. meters (an entire sim).
You can select a land tier in your Account the Second Life website at
https://secondlife.com/account/landfees.php
Please note that under normal circumstances, you will be paying *real* money for your land tier (it is added to your subscription fee) if you choose anything above the bottom (zero) tier.
=== Private Island land maintenance ===
Land on private islands is maintained separately from that of Land on Mainland sims. If you're the owner of the private island (the estate owner), you pay a fixed monthly maintenance to Linden Lab. If you've bought land on a private island from the island owner, then you do not owe a maintenance fee to Linden Lab, but you will have to pay a maintenance fee directly to the island owner (unless the owner is particularly generous).
The owners of private islands currently pay a fee of $295 per month, per island to Linden Lab (as of February, 2007). Educational and non-profit owners pay $150/island. Ownership of a private island doesn't affect your tier fee for any land owned on Mainland sim. Conversely, your tier for Mainland land has no affect on your private island fees. Always check the Official Land Pricing page for the latest rates.
Private island owners set their own fees for people who buy land on their islands. Depending on the needs of the island owner, your tier may be paid either in L$ or by US$ through an external service such as Paypal, directly to the private island owner. Often, these rates will be advertised either on the covenant for the land, or on a website referenced in the covenant. You should always check the covenant for details, and ask the island owner before buying if you can't find the rates or the conditions are unclear.