If you've already been ordered removed (whether you know it or not) then you must expect Immigration & Customs Enforcement (ICE) to arrest you and deport you at any time.
If you're being placed into removal proceedings for the first time, it's going to either be after ICE arrests you, or after you get served with a Notice to Appear (NTA). This is the "charging document" not unlike a criminal summons. It states the reasons the government thinks you don't have the right to be here.
NTA's can be issued by ICE or by USCIS. If you're detained, ICE may also set a bond.
You will then be scheduled for a master calendar hearing before an immigration judge. There are separate dates for those who are detained and those who are not. Your lawyer can file a motion for a bond hearing (if no bond has been set) or a bond redetermination (to try to lower the bond set by ICE).
At your master calendar hearing, you will admit or deny the charges in the NTA, and tell the judge whether you admit you are removable. You will also tell the judge what sort of relief, if any, you can apply for. Or, you can tell the judge you simply want to leave the US at your own expense.
The court will then schedule you for an individual hearing. This is your opportunity to present your case for the relief you are applying for (if any.)
If the judge denies your application for relief, you can file an appeal with the Board of Immigration Appeals, located near our office in Falls Church, Virginia. You don't necessarily get to stay in the country just by filing an appeal.
If the Board denies your appeal too, you may be able to file a "petition for review" with a federal court. If the petition is granted, the federal court will review the case and see if any legal errors have been made.