How to Copy SIM Card Contents to an iPhone
Although your iPhone doesn't store information on the SIM card, if you are transferring to iPhone from another cellular phone, you may have contact information on the old SIM card that you want to transfer.
If your old device is an Android or other non-iPhone smartphone that uses a micro-SIM card, you can typically pop out your new iPhone 4 or 4s's micro-SIM and pop in your old one to copy those contacts to your new iPhone. (Go to Settings→Mail, Contacts, Calendars→Import SIM Contacts.)
Ditto if your old device uses a nano-SIM card and you're moving to a new iPhone 5 or later, which also uses a nano-SIM card.
If your old phone's SIM card isn't the same type of SIM card in your new iPhone, you may still be able to copy any contacts from your old smartphone by connecting it to your computer and using special data export software that may either come with your old phone, or is available to download from the phone manufacturer's website, or by using a third-party smartphone data transfer program.
Check your smartphone's manual or search the manufacturer's support site to see if they offer a software application you can download to export your contacts from your old phone so you can copy them to your new iPhone.
A DIY solution for making your too-big micro-SIM fit in your much smaller nano-SIM slot is to carefully cut the micro-SIM down to nano-SIM size. Search Google on those terms, and you'll turn up a slew of how-to videos on how to do exactly that.
Another option: Check out the nifty third-party program PhoneView for Mac, which offers super-simple drag-and-drop backing up and transferring of contacts (and all sorts of other iPhone data) between old iPhones and new iPhones, and even between iPhones and iPads (handy for transferring your Angry Birds progress between iOS devices so you don't have to replay all those levels you already beat).