What are TLEs?

TLE stands for "Two Line Elements". It is a format for distributing orbital elements data. I'm not aware of where it is officially defined, but several descriptions can be found in the Internet by people that know a lot more about it than myself.

The TLE for a satellite looks something like this:

The first thing that can be noted is that it actually consists of three lines. The first line (line 0) contains the name of the satellite (or other object).
The next lines contain the actual orbital elements. They have 3 common fields:

They start (character 0) with the single digit '1' or '2' to indicate the line number, this works as simple sanity check .
In both lines the next field (characters 2 to 7) is the satellite's catalog number from the US Space Command. The "U" in line 1 is part of the field and it indicates that the elements are for an unclassified object. This is the only kind most of us will ever see.
They both end (character 68) with a single digit checksum. The checksum is a modulo 10 addition of all the characters in the line (including the line number) where letters, blanks and periods count as zero (0) and the minus sign is worth -1. I've read that the '+' sign can sometimes be considered as '0' and sometimes as '+2'.
I found that for this example in line 1 using '+2' for the plus sign works if one assumes that characters 44 and 53 are a '+' sign which is reasonable since they are defined as 8 character long field but only 7 are present (see bellow) AND don't count the minus signs that are part of an exponent. Line two is straightforward.

From here on there are no more fields common to both lines.

The next field in line 1 (characters 9 to 16) is the International Designation. it consists of two digits for the year of the launch, 3 digits for the launch number for that year and up to three characters for the payload ID, since it is common for more than one object to be put in orbit with a single launch. Sometimes orbital debris form the launch are also assigned an ID number.

The following filed (characters 18 to 31) is the Epoch Time, the precise time for which this elements set was generated. This is important since the elements not only describe the orbits' shape and orientation (which change slowly), but also the satellite's position in it. The part to the left of the decimal point is the Epoch's Year (2 digits) and the epoch's day (1 to 365 or 366 for leap years). The number to the right of the point is the fraction of the day starting at midnight UTC. So midnight would be 0.00000000, 12 in the afternoon would be 0.50000000 and one minute before midnight would be .99930555.

The following three fields are drag elements. I don't pretend to be familiar with them. They are not used for the calculations in this script, but should be considered for LEO satellites (not supported in this site). They are mentioned for completeness sake.

Decay Rate (characters 33 to 42): It is the time derivative of the Mean Motion, divided by two (including sign). Sometimes referred to as the "Ballistic Coefficient".
The Seconds Derivative of the Mean Motion divided by 6. (characters 44 to 51). A decimal point is implicit to the left of the first digit and the last two characters are the exponent.
If the SGP4 model was used this field (characters 53 to 60) is the Bstar Drag Parameter or Pseudo Ballistic Coefficient. Otherwise it is the radiation pressure coefficient.

Character No. 62 is the Ephemeris Type. Indicates the model used to calculate the data. If the SGP or SGP4 models was used (almos allways) this field's value is 0 (zero). SGP stands for "Simplified General Perturbations", it is the orbital decay model used by the US Space Command to calculate the orbital elements.
Characters 64 - 67 are the Element Number, a number assigned sequentially when an element set is generated.

Line 2 contains the following elements:

Characters 08 to 15: Inclination in degress. The inclination of the orbital plane relative to the earths ecuatorial plane.
Characters 17 to 24: Right Ascension of the Ascending Node in degress.
Characters 26 to 32: Eccentricity with a preceding decimal point implicit. By how much does the shpae of the orbit deviate from a perfect circle (eccentricity = 0).
Characters 34 to 41: Argument of the Perigee in degrees.

Characters 43 to 50: Mean Anomaly in degress.
Characters 52 to 62: Mean Motion in revolutions per day, an indication of how geosynchronous the satellite actually is.
Characters 63 to 67: Revolution number at the Epoch Time.