When entering an established market where there is at least one other player, there are several ways to acquire substantial market share. To name a few that immediately come to mind:
1 Provide for unaddressed need
Every market represents consumer need and it's a very rare company that can answer every customer's needs.
Example: Google hopes to build a strong cloud-based business providing email services to businesses. However, business MIS takes a very dim view of relinquishing that much control over critical infrastructure. One critical consideration is how secure the comapny's data are in the hands of a third party. One way to compete with Google for this business would be to offer a cloud-based email service where all data were strongly encrypted while in transit and stored on external servers. Google could offer such a service but doing so would remove Google's ability to advertise since they wouldn't be privy to and so would be unable to leverage critical consumer behavior.
2 Develop better emotional engagement
There are so many options here it can and has filled several books. Donate 50% of all revenues to charity, pledge to offset all carbon emissions, provide custom labelled products ("affinity" credit cards), etc. The key here is to identify characteristics with which customers can identify.
3 Integrate with external resources
Build a model that allows customers to integrate the services you provide with those of other providers with which they have an established relationship. Integrate Facebook's profile management with your cloud-based email service to cut out email SPAM, allow users of your microblogging service to integrate with existing geo-presence and mapping services, etc.
Other possibilities: simplify complex features, amplify underpowered features, undercut price, provide better (any) support, etc.
These are all ways to build presence in existing markets but none will be particularly effective without clear, proactive and effective market communications. You can always offer a better mousetrap but if no one knows about your product and why it is better, you're doomed.