If you've never been in a professional role where you hired people, you might have only a fuzzy idea of what happens on the other end once you sent off an application. Here are a few things to keep in mind, based on my own experience:
Hiring managers are busy doing other things. This is perhaps the most important thing you can keep in mind. Except for situations where your application is first screened by an HR person whose sole job is hiring, the person or people making hiring decisions have other full-time jobs and are trying to squeeze this whole hiring process into the midst of their other responsibilities.
This means that despite what you may have heard, "following up" — to make sure your application has been received or to reiterate your interest in the position — is generally going to annoy the person you're e-mailing or calling. Not only have they had to carve time out of their day to review applications, they now have to spend additional time on the phone with you or replying to your e-mail. This is more likely to cast a negative rather than a positive light on you as an applicant.
This is also why it's meaningful to genuinely thank a hiring manager for their time after an interview. Every candidate they interviewed meant another 30+ minutes out of their day when they could have been doing other work. Their time is valuable — acknowledge that.
Your application may not get reviewed right away. Some hiring managers will review applications as they come in, but many have to set aside a block of time once the deadline's passed or enough applications have been received, and this may take a while. This is another reason following up can mean a strike against you — they may not even know if your application's come in because they haven't had time to review them yet, so you risk seeming pushy or impatient if you follow up right away to check on your application.
This doesn't mean you can wait as long as possible to apply, though. It's possible they'll start scheduling interviews well before the listed deadline.
There may not be a concrete hiring timeline. Sometimes a position needs to be filled by a specific date, but often it's not that urgent. A hiring manager may set aside a certain day to review applications but end up pushing it off when an important project suddenly comes up. They may be convening a committee of people to review applications or trying to schedule a panel interview, and it can be difficult to find a time that everyone's free. The fact that you haven't heard anything yet after a week or even a month is not a clear sign that you've been eliminated.
Thus it's not a good idea to try to bug them by asking for a definite hiring timeline or, worse, calling after a certain period of time to ask if you can set up an interview. Only once you're at the interview stage is it appropriate to ask what the next steps are and what their estimated timeline is for making an offer. Even then, understand that this may be a rough estimate.
Hopefully this gives you a clearer picture of what happens after you send off an application. Ultimately it's best to trust that once you're in the applicant pool, a hiring manager is going to move at their own pace and make decisions based on their own criteria. Your responsibility is just to make your application as excellent as possible before you send it.