Employers may advertise opportunities or expect candidates to make speculative applications, depending on the industry. Advertised opportunities should state how they wish you to apply.
Most often, the first stage of application is for you to send a CV and Cover Letter, or an Application Form. If you're successful, you'll be invited to the next stage. For most employers, this is some form of an interview, although some will ask candidates to complete tests or attend an assessment centre.
If you're unsuccessful at any stage, many employers will not contact you to let you know. If you've not heard from them, you can contact them to ask. If you'd like feedback on your application, then it's okay to contact the employer to ask for this. However, not all employers will provide it.
The recruitment process is down to each individual employer. If the employer needs staff immediately then the recruitment process could happen very quickly.
For other organisations, the recruitment process may take longer, perhaps several weeks or even months.