I was dismissed for a breach of confidentiality. I was working on some client confidential information on my client issued laptop and I emailed this info to my personal mailbox as I wanted to continue doing work on my personal laptop; I couldn't take my work laptop away whilst on extended leave overseas. I work within the tech/analytics field. The employer has a policy against this and everyone working there has signed that they read the policy. I just wasn't thinking at the moment I sent the information.

There are no legal ramifications or civil lawsuits at this stage as it wasn't trade secrets or secret IP.

With regards to getting a new job within the software engineering/analytics/data science field, I wouldn't lie on application form and in interviews if asked why I left my old job.

But how do I explain this to show I learnt from my mistake and get a new job ?

Wouldn't employers just throw my application to the bin once I declare I have been dismissed for gross misconduct?

It's all about wording

I think your comment:

The information was work i was working on at the moment and I emailed it as I needed to do work on my personal laptop ; I couldn't take my work station away whilst on extended leave overseas.

Goes a long way to being the right way to describe this. You were wrapped up in a project and yes you messed up but no you didn't mean to. You've learned from this mistake and had no malicious intent.

You violated your contract so your previous employer had little choice but to let you go - your new employer will understand this but if you show them you've accepted responsibility for it and will make sure never to do that again then I think you've got a good chance of getting another position.

There are many ways to say thing like this without lying.
never actually say the words Gross Misconduct. They sound far more serious than what happened.
"I made a dumb mistake and misjudged the sensitivity of some data" is both more accurate and less severe.

Don't say "I was escorted out by armed guards" where you can say "My manager was disappointed enough to let me go"

People have gotten jobs in their field after vastly more serious forkups, don't despair.

If asked specifially try to describe in detail what happened and what you learned from it, for example: ask if the new employer has clear guidelines on data handling.

Yes, some employer will bin you, others might give you a second chance.
You might not immediately get the same job you had before and might have to accept something more junior but be clear in your communications and you'll get there.

Good luck!

Wouldn't employers just throw my application to the bin once I declare I have been dismissed for gross misconduct?

Yes they would, so don't tell them that.

It's hard to answer this question without specifics, but it strikes me as very important to differentiate between an accident or mistake in the sense of "oops, I did that by unintentionally" versus misconduct, as in "this was against policy and I deliberately did it anyways" regardless of whether you knew about the policy or had a good reason to do it or not. What you did was misconduct. You didn't accidentally email the material to yourself, you did it on purpose. Regardless of what word you use when you disclose what happened, understanding that difference, owning up to it, and showing how you've changed as a result is your best hope of gaining future employment.

Passing it off as a mistake, or trying to portray ignorance (in the sense of saying "oh, I didn't realize it was wrong when I did it") is just going to make it sound like you don't bother understanding or following policies. An employee who doesn't know about a policy important enough to fire someone over is just a ticking time bomb to an employer. That's why employers ask employees to sign technology use policies, as it appears yours has - to ensure the employee is aware of policy.

Whether or not you knew about the policy upfront, you need to be ready to discuss steps you take to stay informed about policies and ensure you're following them. You need to be ready to show that you understand that you have responsibility to understand and comply with policy, and that you're willing to do that. That brings us to your questions,

But how do I explain this to show I learnt from my mistake and get a new job ?

First, you need to be able to frame what you did for yourself. Can you explain to us what you learned? How do you approach company policy in general? How to you ensure you're aware of it, and following it?

You certainly don't need to blurt out a 5 minute monologue unprompted, but you do want to be ready to answer these questions because they will come up if you disclose what happened as you intend to. We can't tell you the best way to answer, since the best way to answer is honestly and you're the only one who can give your honest answer.

Wouldn't employers just throw my application to the bin once I declare I have been dismissed for gross misconduct?

Some certainly will, especially those who are more security-conscious. You may want to target less security-sensitive industries or environments until you've worked up enough of a resume after this event to show that you're reliable.

This is mostly a matter of describing your motivation appropriately, and in this respect "At that time, I did not realize" does a better job of conveying that your basis of judgment has changed in the course of that experience than a mere "I did not realize" would. The latter looks more like something that could repeat under similar circumstances.

What you did was a breach of data protection laws in most jurisdictions. Understand this, and approach the question from that angle.

You said it yourself that you were working on client confidential information, and sent it to your personal mailbox.

Businesses have a term for that kind of behaviour, and that is 'data leakage'. In sending that information to your own mailbox, you transmitted the data to a number of machines, any number of which could be intercepting the data for reading, and many do albeit for legit purposes of scanning for advertising relevant stuff or scanning viruses. If any of those connections were being intercepted by an unknown third party, however, you've just put your customer's data into their hands.

Acknowledge that your data security awareness was sorely lacking and that you have worked to fix this in your wording to your prospective employer

This includes understanding what you did wrong and explaining how you might have approached this in future (hint: ask boss, transfer via encrypted USB if necessary and allowed. NEVER by email unless explicitly given the go-ahead). Phrase it as a serious learning point, because you sure as hell aren't going to do it again after getting fired.