A day 2 is any kind of meetup/date/hangout session that occurs after the initial meet/open. This can range from something as simple as coffee, to watching a movie together (either at the theater or at home), to a full-blown dinner with candles and music.
A lot of guys seem to have trouble with what constitutes a good day 2 and some just plain don't have any ideas on what to DO with a girl... so they fall for conventional cliches (like dinner, or a movie at the theater). These CAN work, but they AUTOMATICALLY place you in a disadvantageous position where the GIRL is judging YOU. For example, most girls will tell you that a solid date is a romantic dinner at some fancy restaurant. BUZZ! Wrong. These are pretty shitty dates, ESPECIALLY for a day 2 (which implies that this is the 2nd meet, or the FIRST real "date"). Why? Because it automatically sets you up as, not only conventional, but in an interview-style setup where the two of you are uncomfortably facing each other across a table, all dressed up and waiting for expensive food.
BAD.
The core idea behind good dates is COMFORT. Good day 2's create COMFORT BETWEEN TWO PEOPLE. I don't mean they have to be BORING or UNEXCITING.... NOT AT ALL. However, they are meant to establish a degree of FAMILIARITY and COMFORT between the man and the woman involved... WITHOUT TAKING AWAY FROM ATTRACTION.
You don't want it to be like an interview session, where you bombard each other with questions, blindly groping around to see if there is any common ground. This is how most dates end up, with two people just asking and answering, asking and answering until voila! maybe you hit a common ground! Otherwise.... it falls FLAT.
NO NO NO. Instead, view it as quality time, where the two of you can begin EXPLORING and LEARNING about each other. You want to tease and c/f a bit like you initially did to attract her, but in this case the main goal ISN'T to push her away till she starts chasing you. She's already there. She made it. It's time to show her what you're all about, WHY she should sleep with you. And VICE VERSA (THIS IS IMPORTANT).
Just as you are revealing more about yourself to the girl (and hopefully wowing her as well as establishing comfort), so should SHE be doing the exact same to you. She should want to reveal things about herself to YOU and hope to IMPRESS you.
Good topics of conversation that are open-ended enough to create lots of interesting conversation as well as room for value-demonstration are:
-life goals
-values (what's important to you)
-what you love about life
-powerful/new/interesting experiences you've had
Notice the trend among these: they are not focused on trying to find similarities or "safe" things to talk about, so much as they are mediums through which you can not only share WHO YOU ARE... but ALSO engage her EMOTIONALLY. Girls will get bored if you talk about computer games, pwning on ladder, and why Zerg vs Protoss is imbalanced on Azalea. HOWEVER, they will get INTRIGUED if you focus LESS on the hard data of the game, and more on how COMPETITIVENESS and SUCCESS are IMPORTANT to you, how these shape how you approach your life. You could talk about how any girl you are with has to RESPECT that passion for challenges; how any girl you are with has to share that sense of strength/confidence and be able to back you up against life's challenges.
A minor aspect of day2's is that you can also use this as one final opportunity to really demonstrate some value. Maybe you met this girl at some party where you knew no one and couldn't really show off as much of your social status as possible. You can invite her out with YOUR friends and show off your social connections/popularity. Maybe you have some amazing guitar skills. You can show off by making the day2 at a concert you and your band are playing at.
Here are some solid day 2 ideas:
-cooking dinner together (she brings xyz, you'll have the abc)