I keep reading about adjournments in books, but why don't we use them anymore? They seem like cool analysis sessions.
Primarily because computers have rendered adjournments obsolete. In addition, with the advent of digital clocks with delay or increment, it is much more fair to have people play it out over the board. It is much better to have the players decide it than having a team of people helping them overnight, or nowadays, a strong computer.
Adjournments are still used but very rarely.
They are deprecated and this is shown by the path that the FIDE Laws of Chess have taken over the years in prescribing their use.In the previous incarnation of the laws they were covered in an appendix. In versions before that in the main body. In the discussions preceding the release of the current rules there was a move to remove them but the English Chess Federation argued in favour of retaining them for one more cycle because there was at least one league in London which still used them.
The compromise was that a new section was created in the Laws after the appendices called "Guidelines". The rules governing adjournments were moved to "Guidelines 1". This section is scheduled for retirement in the next release of the rules unless the ECF successfully petition for a further extension.
"Guidelines 3" covers quickplay finishes which is also destined for the dustbin. Both practices made sense before the advent of digital clocks and the use of increment/delay but really have no place in the 21st century because both practices introduce an element of avoidable unfairness. In the age of engines any adjourned game is going to be decided by the engine not the imagination, wit, ability and knowledge of the two players.
Since (fingers crossed) this section is likely to disappear from the rules soon and isn't too long it is perhaps worth reproducing:
Guidelines I. Adjourned games
I.1.1 If a game is not finished at the end of the time prescribed for play, the arbiter shall require the player having the move to ‘seal’ that move. The player must write his move in unambiguous notation on his scoresheet, put his scoresheet and that of his opponent in an envelope, seal the envelope and only then stop the chessclock. Until he has stopped the chessclock the player retains the right to change his sealed move. If, after being told by the arbiter to seal his move, the player makes a move on the chessboard he must write that same move on his scoresheet as his sealed move.
I.1.2 A player having the move who adjourns the game before the end of the playing session shall be considered to have sealed at the nominal time for the end of the session, and his remaining time shall so be recorded.
I.2. The following shall be indicated upon the envelope:
I.2.1 the names of the players,
I.2.2 the position immediately before the sealed move,
I.2.3 the time used by each player,
I.2.4 the name of the player who has sealed the move,
I.2.5 the number of the sealed move,
I.2.6 the offer of a draw, if the proposal is current,
I.2.7 the date, time and venue of resumption of play.
I.3 The arbiter shall check the accuracy of the information on the envelope and is responsible for its safekeeping.
I.4 If a player proposes a draw after his opponent has sealed his move, the offer is valid until the opponent has accepted it or rejected it as in Article 9.1.
I.5 Before the game is to be resumed, the position immediately before the sealed move shall be set up on the chessboard, and the times used by each player when the game was adjourned shall be indicated on the clocks.
I.6 If prior to the resumption the game is agreed drawn, or if one of the players notifies the arbiter that he resigns, the game is concluded.
I.7 The envelope shall be opened only when the player who must reply to the sealed move is present.
I.8 Except in the cases mentioned in Articles 5, 6.9, 9.6 and 9.7, the game is lost by a player whose recording of his sealed move:
I.8.1 is ambiguous, or
I.8.2 is recorded in such a way that its true significance is impossible to establish, or
I.8.3 is illegal.
I.9 If, at the agreed resumption time:
I.9.1 the player having to reply to the sealed move is present, the envelope is opened, the sealed move is made on the chessboard and his clock is started,
I.9.2 the player having to reply to the sealed move is not present, his clock shall be started; on his arrival, he may stop his clock and summon the arbiter; the envelope is then opened and the sealed move is made on the chessboard; his clock is then restarted,
I.9.3 the player who sealed the move is not present, his opponent has the right to record his reply on the scoresheet, seal his scoresheet in a fresh envelope, stop his clock and start the absent player’s clock instead of making his reply in the normal manner; if so, the envelope shall be handed to the arbiter for safekeeping and opened on the absent player’s arrival.
I.10 Any player who arrives at the chessboard after the default time shall lose the game unless the arbiter decides otherwise. However, if the sealed move resulted in the conclusion of the game, that conclusion shall still apply.
I.11 If the regulations of an event specify that the default time is not zero, the following shall apply: If neither player is present initially, the player who has to reply to the sealed move shall lose all the time that elapses until he arrives, unless the regulations of an event specify or the arbiter decides otherwise.
I.12.1 If the envelope containing the sealed move is missing, the game shall continue from the adjourned position, with the clock times recorded at the time of adjournment. If the time used by each player cannot be re-established, the arbiter shall set the clocks. The player who sealed the move shall make the move he states he sealed on the chessboard.
I.12.2 If it is impossible to re-establish the position, the game shall be annulled and a new game shall be played.
I.13 If, upon resumption of the game, either player points out before making his first move that the time used has been incorrectly indicated on either clock, the error must be corrected. If the error is not then established the game shall continue without correction unless the arbiter decides otherwise.
I.14 The duration of each resumption session shall be controlled by the arbiter’s timepiece. The starting time shall be announced in advance.
The question doesn't indicate what context it's being asked in, but at least in the US (in USCF events), adjournments are generally not used because most tournaments are played using sudden-death time controls. This makes it easy to predict (at least approximately) when a round will end, because a player can't get an additional time period by completing a certain number of moves. Once the time period expires in a sudden-death time control, the game is over, regardless of how many moves have been made.