That gov.uk link is a very good one. There are some non-government sites which also give advice on running a small business and others which have templates for invoices. This Google Doc lists all the links I wanted to put in this post, but couldn't.
However, I recommend you use some business accounting software. When I started out freelancing, I used Word and Excel to make invoices. But when it came time to file my tax return, the records I'd kept were inadequate. I now use a web-based software that records my invoices, business expenses and income. Some are listed in the Google Doc.
There is no difference between a "VAT invoice" and a "non-VAT invoice", except the latter doesn't include line items for VAT. Just tell your client you're not VAT registered if they ask.
You don't need to give your employer your NI number in your invoice. However, they might ask you for your UTR – unique tax reference. Clients don't need it – I suspect they use it as proof you exist, in case they get audited.
I'd recommend you get this book on business accounting- and if you plan on freelancing full-time, find an accountant. Search around for one who works with other people in your field, and ask them how much it'll cost do to your taxes. I guarantee a good accountant will save you money.
BTW - If you haven't already, you need to register with HMRC as being self-employed. And as soon a client pays you, put 20% of that money into a savings account or an ISA (preferably one that doesn't give you instant access), so it's set aside for when you have to pay your tax bill. I didn't do this when I went full-time freelance a few years back and I'm still paying the price for it. (Literally. Every month. To HMRC.)