The basic op-amp has inverting and non-inverting inputs, and an output. There are also 2 power supply connections. These are meant to be a dual power supply (centre tapped). The centre tap provides a common reference point of 0V. External connections and measurements can refer to this 0V point.
Inverting means the output is the opposite polarity to the input, inverted or upside down.
The usual op-amp is a voltage amplifier with differential inputs. The inputs respond to the voltage difference between two points (known as differential mode), while ignoring the voltage between them and the power supply and common (known as common mode). If the + input is more positive than the minus input, the output swings in the + direction. This is non-inverting. If the + input is more negative than the minus input, the output swings in the - direction. This is inverting.
The voltage gain is the ratio between output voltage and input voltage. If 1mV difference between the inputs causes a change in output of 1V the gain is 1000 times. This is often stated as a logarithmic ratio, 60dB in this case, as 20(log(ratio)) because it is a voltage ratio. Real op amps might have a gain as much as 120dB = 1 million times, so only 1uV causes 1V change in the output.
The gain is very high, so a few microvolts can drive the output to full scale (saturated near one of the supply rails). This makes the op-amp on its own difficult to measure, as its output is usually saturated in the + or - direction. Consider several microvolts exists between two pieces of wire touching (dissimilar metals, thermocouple effects). However its purpose is to act as an "operational" element in an overall circuit
The ideal op-amp is just an explanatory model. In practice real op-amps can approach this, and data sheets are important towards understanding the limitations. The 741 was one of the first available easy to use devices, so it is a suitable benchmark to compare with others, all of which are improved in some way. Here is a basic data sheet. It pays to understand every single specification on this if you want to use op-amps much.
http://www.ti.com/lit/ds/symlink/lm741.pdf
The typical basic operation is as a non inverting amplifier with a gain of 1. This is called a unity gain buffer or a voltage follower (diagram below). Its purpose is to "buffer signals". That means what happens at the output has little effect on the inputs. More or less current can be drawn without changing the voltage from the input circuit. It changes a high impedance source to a low impedance source.
The signal is between the plus input and the 0V common. The output is measured between the output and the 0V common. To understand op-amp circuits, most of them rely on negative feedback that strives to make the voltage between + and - inputs close to zero. The output has a feedback connection to the - input. The input is a differential amplifier, true, but acts as an analogue comparator, meaning it compares the two signals and produces a difference, which is the error x gain. If the output is too high (compared to the signal) the error drives the output in the negative direction. If too low it is driven in a positive direction. This makes the difference between output and signal less, until there is little error. The output settles where it is almost the same as the input, and the error is just enough to keep the output at that point. As the gain is very high, the error is nominally very small (maybe 1/millionth of the signal voltage). The difference is the error, which is amplified and becomes the output.
Other circuits can be understood in the same way, except that the feedback path is modified to achieve some operation. For an amplifier with gain, the output fed back is reduced (divided by resistors) so more output voltage is needed to match the input signal.