Institutional traders trade both for clients and with their own money they trade both the same way but the traders book them to different accounts when their orders get filled. When they trade with their own money we call this proprietary (or "prop") trading. When engaging in prop trading many institutional traders also act as "market makers" where they get paid for providing liquidity (i.e. fulfilling open market orders). I once worked as a developer on two of the tools or "Order Management Systems" (OMSs) which they use to do this. Basically these tools allow the traders to enter, amend and cancel orders, view their current positions (how much of each instrument they own), book completed trades to their accounting system, and view various graphs and statistics on the instruments that they are trading in. The OMS sends any orders directly to the market or to systemic internalisers who complete trades for their clients.
Large orders tend to go through either smart order routing or some type of algorithm (algo) before being sent to market. These both do similar things; they break up the order quantity such that the volume traded does not have too significant an impact on the price either by sending the broken down volume to different venues on the same market or by spacing the order volume out over time. Many of the algos used specifically target keeping average price to a minimum. Dark pools, where orders sitting on the order book are not published, are also a way of preventing traders from knowing your order size and so moving the price. The institution's trader will have a great deal of discretion over which of these methods are used.
Once the orders are complete they go directly to the market rather than through a broker and are fulfilled by other market participants.
The amount of an instrument (equity, future, option commodity etc.) that they can buy in one day will be governed by a number of things, most notably how much cash or credit they have (they normally have more cash and cash equivalents on hand than most human beings will see in their life), how much they can afford to move the market price (including how fair they think the valuation is currently) and the liquidity of the market for the instrument as a whole. If there is little daily volume and little interest other than the institution it may not be possible to buy 100 let alone 1M shares.
On very rare occasions it is possible to do an over the counter deal with another large holder of the instrument but this is almost always illegal now as legislatures make this opaque deals harder and harder to justify. The only real case here these can occur in modern trading is where an institutional trader has already declared a takeover of the company and made an offer to all of the shareholders. Even the rather arcane fixed income markets (bonds) are being forced to move to transparent electronic platforms from direct phone trading to prevent market abuse.
note: I've not gone into full details of anything here so check for other people's answered questions on areas first then ask further questions if you have any.